Am I making a big deal out of nothing? Should I just ignore it or try to cure this low-level hum?
You've done an excellent job troubleshooting the problem, and having everything plugged into a UPS powered from the same outlet eliminates the possibility of a ground loop. Clearly, the DVR is at faultit's probably a minor grounding problem within the unit. If you can't hear the hum when watching TV, I don't think it's a big deal, and I'd probably ignore it. Otherwise, the only solution is to contact Dish and have them swap the DVR for another unit.
If you have an A/V question, please send it to firstname.lastname@example.org.