Yes, I suspect you're right. Most 230V in / 230V out VFDs don't check for phase voltages, 'cos it really doesn't matter to them. However, the type that boost 230V in to 415V out have front end which might get upset if a phase were missing if they use a 3-phase boost power factor corrector (PFC) front end.

I've avoided the need for one of those 230-415 VFDs but it must be possible to find some that DO run on single phase because I've seen a few recommended on various forums.
Sounds as if Jazz knows of some examples.

A simpler solution would be to either modify your existing motor for 230V delta connection or replace it with a 230V motor. Then you can use almost any 230V VFD. That may work out cheaper in the long run, as a boost VFD is going to cost you more.