BAFA funding confirmation - Execution?

Hello,
I had decided on a construction company, but had not signed anything yet, since the approval planning was done independently with an architect.
There were some problems, and everything was delayed by over a year and had to be replanned.
The construction company has now significantly increased their prices, but the fact that the house is now also smaller was (in my opinion) not taken into account, because then the price should have come out approximately the same as before.
There were also problems with the KfW confirmation. I waited 3 months for the additional cost from KfW55 to 40+, and when I wanted the confirmation for the bank, they said the energy consultant had made a mistake and now everything was more expensive and complicated.
Therefore, I have now obtained new offers from two other companies.
If the original company still can’t manage it (I was supposed to sign this year but still don’t have the new price...), I would consider changing the company after all.
Now the advisor of one company told me that his heating installer said that the company that submitted the application to BAFA must also install the heating system. Is that really the case? I can’t imagine that. Because I am still allowed to change the heating, right?
Has anyone had their application submitted by one company but had the heating installed by someone else, and did that cause any problems with the funding?

A look at the BAFA website helps. You can install anything as long as it is of the requested type and also eligible for funding. You just won’t get more money without submitting a new application.

Thank you for letting me change the heating system again, I am aware of that. But what the construction company says is that the company that submitted the application must carry out the work. However, can I also have another company do it? After all, I am the applicant even if someone else did it for me. I cannot imagine that you are bound to a company just because of the application submission.