Bell Services is designed as an amenity for guests of that hotel so the basic answer is no. Whenever we've used them, we've been asked if we're guests of the resort. You might get lucky and no one checks your room number when you drop them off, we don't really know what they do once they take them. This also feels like a security issue for me. If just anyone can drop off a bag, that's not good.
I get wanting to check out a restaurant. Even when staying on property, it's a pain to have to go all the way back across WDW just to pick up bags. It's an extra Lyft or whatever, but your best bet is to store them with your Universal hotel, ride over and enjoy, then stop and pick up bags on your way back.