As always, baking has dangers, and all the components would prefer not to be subjected to those kinds of temperatures.
The bigger deal is that a power supply that doesn't work is very probably a result of something other than a bad solder joint. If it turns on barely, that probably means the whole circuit is connected, so it's not a bad solder joint. So baking won't fix whatever is wrong with it.
If, for example, some capacitors are blown and the power supply can't operate stably because they're running so out of spec, heating the caps up even more is the opposite of what you want to do.
Have you confirmed the power supply doesn't work in general? Tried in another system? Tried with less things plugged in? Tried to jump start without connected to motherboard, via green+black wire paper clip test? And so on.
You should just get an RMA on the PSU if it's under warranty, or otherwise replace it.