Yes, the UPDATE statements could also be slow, but the INSERT statements may cause page splits etc, why they are more prone to stalls.

No, TVPs to not persist between calls, but nor is there any reason to. Anyway, that discussion is immaterial for you, since you cannot change the application. (But you could kick the vendor in the rear parts for not cleaning up his act.)

Anyway, it seems clear that there is an issue with your mirror or the connection to it. That is, either the network connection to the mirror is flaky or has too little bandwidth, or the mirror has problems with the I/O subsystems. The ping times you noted hints that the network is the problem. 25 ms is far too long for a LAN connection. But it seems you have a difficult battle with your infrastructure guys.