Reflow of the ground pad for the FPGA didn’t improve the situation. All of the pins on the FPGA have a super solid connection to the PCB. However, there was a single loose pin in U1 and about 12 loose ones on U3. I’ve reflowed those, so it’s off to play some games and see how long it lasts until power blinkage this time. Fingers crossed it’s a few years this time again.
EDIT: 2 hours of gaming and no dropouts yet. I’m considering this fixed, until a longer gaming session potentially proves otherwise.