What can I do to thicken the mixture in a meat "cup?"
A little cornstarch should, if you add it with the bbq sauce, make it just thick enough to cling together a little better.

I'd say make a slurry of say 1/2c water, 1/4c bbq sauce, 1tsp cornstarch and add it to your ground beef. Let it cook down (I'm basically thinking how taco packets work here) until it's close to where you want it; continued heat in the oven should finish binding.

Or skip the bbq sauce, use dry herbs and spices mostly with just a tiny dab to bind it together.

Also you just made me want a patty, you awful man.