Such damages are occasionally seen where both components belong to the same owner. For the initial diagnosis, a roofer is sent out, which is usually sufficient. Common causes are a leaking seal and a clogged drain. Both parties could initially share the roofer amicably; perhaps that alone is enough.
I would not advance payment with a specialist; I think it is too early. The other party wants something and can therefore initially cover the costs themselves, especially since a court would follow their newly commissioned neutral expert.
It is certainly advantageous if the "accused" signals that he/she will of course comply with an undisputed obligation.
The dew point is more important than it initially sounds. It is enough if the affected room is too cold; nothing else has to be broken.
Best regards
Gabriele