Cost coverage for fencing that is not desired

And so quickly you can mentally annoy your new neighbor..

It's all a bit contradictory.

According to the Hessian Neighbor Law

. . you are obliged to cover half. You have read and know the Hessian neighbor law. You are aware of it.
And yet you write something about YOUR ideas,

My previous idea was that I would do it on OUR property at MY expense.

That does not fit together.

And then you backtrack a bit: you write once that you have no capacities of any kind yet to take care of a fence/garden design. Then you write that you want something green. Then you write that you want a privacy screen plus hedge, for which you would have already had L-stones installed. In my opinion, these are defensive arguments because you cannot do as you want.

Then it turns from "neighbor approached us" into "being forced at gunpoint," then you talk about submission. Is there really no normal coexistence for you, where one asks, the other answers, you first listen, chat, and agree on material and schedule? Your neighbor law at least still provides the tolerable compromise alternative of a standard fence at 120 cm height if you cannot agree. So you cannot be that poor after all.

I do not believe that he accepts anything other than the fence he wants on the other 3 sides anyway.

So far that is your assumption. One reads that it is actually you who does not want to make any restrictions.
In principle, one can very well vary one’s own property boundaries in many ways and diversely.

Can I actually demand in return

So far no one is demanding anything that they may not or cannot..

I find nothing on time limits

Something like that is generally regulated by common sense. That actually plays into your hands. The built-upon or used property counts after all. In my opinion, the house would have to be more or less ready for occupancy. From there counting starts. Because it applies that:

Your house is still a shell but is still under construction, so legally it counts as not yet built in my understanding.
However, one can understand your finished neighbor.
The order is arrival, request, sitting together, ordering... Overstraining anything serves no one.

If I were you, I would reset a bit and try to talk to the neighbor on equal terms. He asks, you can calmly accept a conversation without immediately thinking of a duel. You can gladly leave your weapon at home and instead provide some beer.
And as in a marriage, it usually works better if you briefly put your own wishes aside to listen in order to possibly enforce your own wishes. Also not having to be constantly asked belongs to that. However, you can appeal to his reason and ask if the installation can wait until your earthworks are leveled after the rainwater connection.

You can also throw something into the round: what is the maximum value for boundary fences in your area? 180 cm or really only these 120 cm?
He must take your height into account and may not put a man-high fence right in front of your nose.

There is a risk in something like this of following Paul Watzlawick's "Anleitung zum Unglücksein" and we are all at least somewhat affected by that. It's easy to get the wrong idea about a word and before you know it, there's a sullen mood, also because everyone is understandably annoyed during the construction period.

oops... saved too early.
The neighbor can and should also bring his ideas to you. I think that’s better than if he just does something and you then wonder why he didn’t ask. Besides, with his own fence, he would have to stay a bit on his side, and you too.
Of course, it may be that he has strange or self-serving views, which is not uncommon. But only then can you tell him that you might have an ordinary fence, maybe chain-link or a simple type from the landscaper, but since you will have beautiful hedges or similar plants all along, the fence won’t really be seen anyway, so you want to spend as little money as possible on it. As far as you know, a fence of [xy] height is required, so let’s make one like that together.
It was similar for us and we didn’t want any fence at all on our large property. But with that, we are almost the only ones here. The neighbor wanted to put his fence on the boundary even before we built our house because they have a dog. I wasn’t in the mood for it back then either, but with a manageable semi-detached house, it’s something different.
We solved it here by each taking a part. My landscaper set the curbs and the posts, the neighbor got the fence panels, and we also dug some together. I don’t know if it was equal financially or effort-wise, but he was satisfied and for me it was okay even if I didn’t really need a fence. Now the neighbor’s property is bordered, and the rest around it is completely open.
I also had that feeling of being overridden back then, also because if I had built a fence it would have been a neat wooden one, but they didn’t want that because around it there are also wire mesh panels.
The landscaper can also put in simple sleeves, and you build something out of wood, maybe even together. If he notices that you are making a move towards him, he might do the same. You should/must try and not expect something negative right away. If he really acts unpleasantly, you can say that you agree to follow the applicable rules and implement what is generally required. A call to the building authority of the municipality should clear up any uncertainty regarding fence regulations.

Basically, I can understand that he wants to sit in his garden in the summer. Well, he’s done that pretty well so far and could do the same this year, but we’re now on one side and the most important thing is a fence first! So, I don’t know, but my idea of living nicely together is not annoying the new neighbors right away with a fence (by the way, that has something like “I actually don’t want to have anything to do with you”): I would have wished he would come over, say hello kindly, and after some small talk say “And at some point, we need to talk about the fence and what we want to do there. But first, come on over.” At least that’s my naive idea...

But unfortunately no, to make it clear I am addressed with the wording: “We have now commissioned a landscape gardener to fence the property. It’s customary to split the costs in half, will you take over your share then?”
Yeah, what are we even talking about? Is there an offer? Yes, but for everything? He can’t tell us the cost for the middle part. And when I asked if a hedge could be put in instead, he said yes, behind the fence on your property, he doesn’t want a hedge. So much for rigid views. And that is exactly not my idea of “chatting about it over a beer.”

And then you backpedal a bit: You write at one point that you have no capacity of any kind yet to deal with a fence/garden design. Then you write you want something green. Then you write you want privacy plus a hedge, for which you already had L-shaped stones installed. In my opinion, these are defensive arguments because you can’t do what you want.

Well, having an idea is a long way from implementation and of course we need to think ahead on the groundwork. Setting L-shaped stones afterwards is quite difficult. It’s 3m at the house to screw the privacy screen onto. Behind that, you “could” plant a lower hedge (or fence or whatever). L-shaped stones and an idea are not yet a garden plan.

Then from “neighbor spoke to us” it turns into “pistol at the head,” then you talk about submission. Isn’t there simply normal cooperation for you, where one asks, the other answers, you listen first, chat, and agree on material and schedule? Your neighbor law at least allows the tolerable compromise alternative of a standard fence at 120 cm height if no agreement can be reached. So you really can’t become poor.

So far that is your assumption. Reading it, it’s actually you who doesn’t want to make any compromises.
In principle, you can very well vary your own property boundaries in a versatile and varied way.

Okay, I’ll repeat myself: how do you feel about these sentences: “We have now commissioned a landscape gardener to fence the property. It’s customary to split the costs in half, will you take over your share then?”

“We were thinking of a hedge or something green as a divider?”
“You can do that behind the border fence on your side.”

Stuff like that is generally governed by common sense. That however plays into your cards. It is about the built-on or used property. In my opinion, the house would have to be more or less ready to move into. From then on, the counting begins. Because it applies:
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Your house is still a shell and still under construction, so legally it counts as not yet built.
However, one can understand your finished neighbor.
It’s about arriving, asking, sitting together, ordering… Overstraining anything doesn’t help anyone.

…
You can however consider appealing to his reason and ask whether the installation can wait until your ground works are leveled after the rainwater connection.

He has already commissioned the landscape gardener and wants to start NOW, nothing about arriving and the house is not move-in ready yet, etc. Honestly, if I quote paragraphs of the neighbor relations law to him that there is still time, it doesn’t improve the mood.

I only see someone who is overstraining things and causing stress. It’s not like he can’t sit in a nice garden right now....

Sure, it may be that he has strange or self-serving views, which is not necessarily uncommon. But then, and only then, can you tell him that you might have a normal fence, perhaps chain link or a simple type from the gardener, but since you will be planting beautiful hedges or similar plants all along, the fence won’t be visible anyway, so you want to spend as little money as possible on it. As far as you know, a fence with a height of xy is required, so let's make one like that together.

...

I also felt a bit run over back then, also because if anything, I would have built a nice wooden fence but they didn’t want that because the surroundings also have welded wire mesh fences.

Thank you for your response, I think it will be similar here then. I could also imagine a nice wooden fence or WPC, then no hedges would be needed to look nice :) But of course that is a different financial level than a standard welded wire fence. We’ll see what his garden landscaper has to offer.

You will live alongside him for the next few years. I wouldn’t make a fuss but rather approach it constructively. And maybe also mention that you are currently not ready. Perhaps he is also understanding.