I own a small ultrasonic cleaner I use for cleaning technical drawing pens but most are sold as a way to clean jewelry. Seems to me if you had a clean unit (never had a solvent used in it) put in some distilled water, let it run for 15 minutes or so... it just might be a perfect answer.
I've forgotten I was running mine a couple of times and came back to find it empty of solution, so It gets plenty warm enough to evaporate off the soapish cleaning fluids I use, but nowhere near boiling. I've a feeling if I tried boiling temps on plastics, I'd wander off spacing out and come back to a warped carto. I am way too drifty.
I've no idea if the fiber material could withstand the sound waves or not. But it is harmless to even frail metals and only made my plastic pen barrels sparkle... never distorted their shape. Natch, it would just have to be an experiment on an old cart you didn't care about the first time.
Genie