Probably going to be easier than expected. I popped open the cover to see whether it was like the much older Netcomm switch we have.
Sure enough, Linksys do it the same way in the LGS326AU:
This PSU is an open-frame PSU made by Asian Power Devices, Inc. Model NW-20A12-BAAB. Not sure if there’s someone here who knows more about this particular PSU.
It appears it’s the one 12V feed split in two. Red/Black are negative, green/yellow are positive. I’ll double check this. I think with a nice big inductor/capacitor as an in-line filter, this can be hooked straight up to the 12V line, perhaps with a small amount of zener overvoltage protection. It appears that the voltage is further rectified on the switch mainboard: