After repositioning all the relevant joints to be inside the model, I tried playing around with the "max influences" setting, which changed the amount of distortion, but did not end it. I'm certain that the elbow joint is properly bound, as I can pose the model without trouble before adding the influence object. It's not that important to my project, just bugs me to see those squashed elbows.
It now looks like this (max influences set to 3):