What to do?
@roland is right: contact the local Mieterverein. The Mietervereine are consumers' rights organizations that give legal advise on rental matters to renters.
This is usually the cheapest option to get a lawyer who is specialized at rental law.
The Mieterverein, btw., may find that the final accounting is wrong - the rules are rather complex (they have regular press releases that many or even most such accounts are wrong).
Nebenkosten/service charges, running costs...
source Default mode for Nebenkosten is to pay monthly advances and then have an annual invoice for the precise amount. It is possible to agree on a contract that uses a flat rate fee instead, i.e. there is no such final accounting and invoice. In my limited experience, that is cusomary for shorter-term rental agreements (say, subletting or renting for a few months where it is difficult to do the final invoice within a reasonable time) but unusual for rental contracts that are supposed to last over several years. The linked post says that if the contract is worded ambiguously, a proper annual and final accounting is to be done.
IANAL, but my lay understanding of your contract is that the text is at the very least ambiguous: "pauschal" points to flat rate, but a flat rate must be an exact fee, not "up to" (bis zu). Moreover the wording is ambiguous whether the up to refers to the running costs being covered up to the specified amount (i.e. everything above that you have to pay extra, likely the intention) or whether the running costs are capped at the specified amount (i.e. can amount to up to x).
According to this other post a contract may have some parts of the Nebenkosten as flat rate and others not. They also say that some specific running costs like heating and warm water must get a proper accounting.
Thus, I'd expect that in the end, getting a final account and invoice is probably correct since the attempt to make it a flat rate (if it was one) is likely void. (Final account is needed regardless of whether you have to pay or you have to get back some money).