I tried soldering and it didn't work, the console still reset during play. I found a spare PSU for sale on etsy for cheap enough that I didn't mind just paying for a replacement.

Funny thing is, though, after I soldered the old PSU I decided to test it a bit by playing Symphony of the Night. During the intro with Richter during a couple of screen transitions, what almost looked like a BIOS screen would pop up saying "OVER: 80191" and some random number strings. The music kept playing, and when I hit start they'd go away and the game would be right where it was. It happened twice and was really weird. I Googled it but nothing came up. I know it was just a coincidence, but it freaked me out!