So, I'm posting this just in case it helps anybody else out:
I was having problems with the webserver crashing on my duet Ethernet. People here suggested all sorts of useful things. Some seemed to help a bit, some didn't. But the problem never entirely went away...
Well! I finally figured it out: Because all my steppers are mounted on shock mounts (Voron2, if anyone is interested), their casings are all electrically isolated. And there are lots of belts in my printer. So apparently, belt drives + isolated steppers = static build up. And apparently, the first consequence of that buildup is crashing the Ethernet module.... So, I added grounding leads to all my steppers, and Boom! total stability.