Ventilation system as a combined unit with Bafa funding

Hello everyone,
slowly we have digested the shock of the KFW funding stop and have shrunk the equipment/size back into the budget. The controlled residential ventilation has also been dropped. Our carpenter says his wall construction doesn’t need ventilation, and that’s why we crossed it out.
We still have a joker with the BAFA funding from 2020 which was extended for us once more. With a combination device like Tecalor THZ we would get just under a third funded. They would also be delivered to us in time. All other heat pumps only we will very probably get only after the deadline and I would have to ask BAFA for an extension again.
We have an offer and we would have to invest just under 9k € ourselves for it. Would you do that? Is that nowadays part of the equipment? With the manageable additional costs a nobrainer?

Our carpenter says his wall construction does not need ventilation

Which he lied about. He is not allowed to build a non-tight wall. I cannot contribute anything to the rest.

That was certainly not meant that way. Of course, you still have to ventilate 1-2 times a day to have enough oxygen to breathe. He just meant that due to the Fermacell and the wall construction made entirely of wood, a lot of moisture is buffered and mold is almost ruled out.

So he meant "My wall structure is hardly susceptible to mold."

That is already a difference and only one (in new buildings quite unimportant) aspect of ventilation. So no matter what you do now, try to get active ventilation into the house. The simplest, cheapest (and not necessarily worst) option would be small (80mm, 100mm) exhaust fans in the wet and utility rooms and, if necessary, retrofitted window frame vents. That doesn't provide heat recovery, but the additional costs over decades are likely to be less than the ventilation you saved.