But it absolutely *will* detect this. And event if it somehow doesn't, remember that inverters are a. current sources and b. have hard voltage limits.
Remove the load by unplugging and the current will immediately drop to zero and the voltage rise to beyond the limit voltage. Redundancy...
regardless of it being room-sealed I don't believe you can have a woodstove in a room which has an extractor of any kind. Read the B-Regs very closely for your jurisdiction...