I was starting to think that I need to go to a higher-voltage MOSFET. I've been looking on Digikey particularly at the R6530 family of MOSFETs from Rohm - 650V, 30A. There are small variations that I don't quite know how to choose between. For example, the R6530KNZ4C13 is described as a "high speed switch" version, while the R6530ENZ4C13 is described as a "low noise power" version. The specs seem nearly identical to me, with slight variations in total gate charge, rise or fall time, etc. I don't know how to judge whether or which would be better for my needs.
I've also wondered whether an IGBT would have any advantages for this purpose. My *extremely limited* understanding, which may well be incorrect, is that "in general" MOSFETs can handle higher switching speeds, while IGBTs can handle higher currents - but of course, when one starts looking at the hundreds or thousands of different options, there are plenty of what seem to be high-speed IGBTs and high-current MOSFETs. One thing that is distinctly different, from what I can tell, is the voltage drop across the IGBT C-E - a couple of volts - vs across the MOSFET D-S - only millivolts. Not sure if that makes any practical difference in my application.
I'm learning so much ... and yet the more I learn, the more I recognize that there is so much more to learn!