Is our new building realistically financeable?

Have you saved the 240k for the apartment in your hometown and the currently available 50k yourselves? Or were there family subsidies?
Overall, the 240k for the apartment is simply a bad investment given the rental income, which is now unnecessarily complicating your house construction.
You can get 140 sqm without a basement for less than 400k. Therefore, I still see possibilities for you.
You still have over 10k annually in reserve for other savings (13.5 + 14 salaries), which would make the whole thing possible for me.
6700 EUR/month with a 2500 EUR installment + at least 10k in "additional salaries" annually sounds not bad to me.
As I said, not many have 4k left after the house installment. And if you saved the 240k for the apartment yourselves, you have proven that you can budget well.

well you saved and invested well and then the apartment came. Why do you buy an apartment for 240k in cash? To go on vacation there? Probably not to get 340 rent per month.

Now it gets more difficult with home financing. and "gentle" hillside and architect's house is for me an open-end investment... that will be more expensive than the 600k

If you justify the house as an investment, you should also apply the standards of an investment to it. And by that measure, a property performs relatively poorly, especially in terms of concentration risk and opportunity costs, but also in ongoing costs. You can save without property too, you just have to set up a standing order that removes the money from the current account on the 1st of the month. That doesn't mean you shouldn't buy/build a house, but you should be aware that it is not necessarily justifiable as an investment. Especially if at least the builder would also be happy with an apartment ;)

It’s not always just black or white. Here it was argued that the decision should now be made so that one is eventually carried out feet first. In response, I like to offer the idea that this is probably not necessary. Depending on the area, real estate has achieved excellent returns in recent decades. That doesn’t have to mean that you buy the house solely for return reasons and that there are no better investment opportunities. When I look at the beautiful life that the now 60 to 70-year-old sellers of their 40-year-old houses in the Düsseldorf surrounding area can enjoy: I can only congratulate them. It is almost irrelevant how much outstanding debt they may still have on their property. In any case, it is only a fraction of the selling price. Of course, no one knows if it will look the same in 20 years. And of course, what was written before also applies to apartments.

As said, not many people have 4k after the mortgage payment.

I wouldn't necessarily agree with that. You just shouldn't be dazzled by the income. Either you have saved 150-200k€ in equity (which is not impossible at 42) or you build cheaper and have a smaller installment. That way, you also end up with 4k€ after deducting the installment plus additional costs. That is not necessarily unusual.

Little time + high loan + no equity = high installment

I wouldn't sign off on that now. You just shouldn't be blinded by income. Either you have saved 150-200k€ in equity (which is not unusual at 42) or you build cheaper and have a smaller installment. That way you still have 4k€ left after deducting the installment including additional costs. That is not necessarily unusual.

Little time + high loan + no equity = high installment

I don't know. Without a house, I wouldn't have saved 240k at 42 :) rather spent it

don't know. without a house I wouldn't have saved 240k by 42 :) rather consumed

I can see we understand each other. But surely you wouldn't have blown 240k€.