"I'm confused about how I then get the paper towel I have been holding on the push bar into the bin afterwards. The lids snap viciously and child-severingly shut the moment you release the pressure on the bar."

Oh, ick. I was thinking of the kind of bins we have in this area, which close very, very, much slower. Your building management seems like they're actively working to create more disabled people, as well as being hostile to those already using their facility. (I have limited use of my hands. It's a sore point, so to speak.) If I were you, I'd be really tempted to just litter.