I think the majority of the PSU components should be okay in the ultrasonic bath --it's a bit tricky to find quality ultrasonic cleaning information online, as unfortunately, most of it is junk. From what I've read, the main concern is damaging delicate MEMs components on much more modern PCBs (like on an iPhone logic board). The component that looks like a roll of tape is an electrical transformer, and it may not like getting wet. Something I've seen people say about ultrasonic cleaners is that if you shake anything hard enough for a long time, it will eventually break or degrade.
I wouldn't clean the PCB for longer than a minute or two, as you'll risk damaging things the longer you run the machine, but it is probably fine for a short time. What kind of cleaning product are you adding to the water? I have had good results with a chemical called Branson EC mixed with distilled water.
After you clean the PCB, you'll need to make sure it is completely dry before applying AC power, as residual moisture could be really risky to have around in this case. I think if you stick the PCB in the oven at a low temperature (nothing near hot enough to melt plastic) for a while, the water should all evaporate.
As an alternative (and less risky) solution, you could purchase enough isopropyl alcohol to give the entire PCB a bath. The isopropanol should dissolve the flux residue fairly well. A toothbrush works well to agitate the flux, and paper towels would absorb most of the isopropyl alcohol after you pull the board from the liquid. I think that this technique should be much faster than swabbing the entire board with Q-tips.