All outlets on one 15-amp circuit, need AC unit
We resolved the problem with the portable AC by purchasing a heavy-duty, 15-amp, 12-gauge extension cord and running it from the bathroom to the AC by the window. Since the bathroom is on its own circuit, the AC does not trip the breaker. A big thank you to those who suggested this in the comments.

Ironically, management has decided to replace our central air conditioner after I discovered mold in the unit, and their subsequent inspection also revealed it to have a serious Freon leak.