VFD intermittently trips breaker on bypass relay

[m_c]

I don't quite understand why the contactor coil would burn out, as that should be totally independent of anything that the VFD is doing.

It does sound like an issue with the VFD though.
I've never really paid much attention, but I'm guessing the internal relay only powers up once the capacitors have charged, to ensure the control circuitry powers up cleanly.
Something like a shorted Mosfe/IGBT/<whatever power switching semiconductor is being use> would cause a trip at that point.

I'd try what routercnc suggest, and power it directly, but also disconnect any other wiring to eliminate any external issues.