Gravel bed over property boundary - is approval necessary?

I assumed, as I understood your text, that the top edge of both base slabs is at the same height.

From my point of view, this excludes option 3.

I would handle it very pragmatically. Have the consent declaration given to you as a Word document. Then you add (or have your construction lawyer add) that all additional costs for you (shoring, special measures, or whatever) are to be borne by the neighbor. Perhaps just enter a flat rate of max €50,000. And the matter will be settled...

Please no.
If the neighbor does not shy away from conflict, he will find another solution and you will have to shore up. It has to be resolved differently somehow.

please no. If the neighbor does not shy away from conflict, he will find another solution and you have to intervene. It has to be resolved somehow differently.

You don't really have to do anything. You should. But that involves two parties. One is currently willing, the other rather not? If the neighbor is in such a huge hurry, he will not find a suitable solution anytime soon. And if he does find one, but the OP cannot keep up quickly enough, then so be it (bad, but you have to live with it). The OP can't prevent that anyway if the neighbor is not cooperative. ==> and it doesn't look cooperative at the moment, otherwise he wouldn't press so much for consent. And the OP should definitely not give that lightly.