I notice Flank steak already made the list, and once I would have agreed, but it's gotten trendy, and with the trendiness, expensive. It's still not fillet-price, but it tends to run in the 7-8 dollar a pound range where I live.
The steak I like to grill that I find to be tasty, cheap, and available is skirt steak. Do a light marinade, grill it as lightly as possible, cut across the grain of the meat, serve.
Your mileage may vary, however. The cheapest cuts are almost always the ones that local cooks don't know what to do with, and if the locals are hounds for the fajita, you're not going to be able to find skirt steak at all.
My advice is to go to the store with an open mind, and browse the meat counter looking for deals. I came home with 3 pounds of tenderloin tips for less than twenty bucks the other day: if I'd gone out looking to buy such a thing, I'd have been disappointed.
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.
Best Answer
From a food safety point of view, no. There is no danger, because the meat contains no pathogens after overcooking.
From a "healthy living" point of view, it might be a problem, because you can have created carcinogens by charring. But we don't discuss such topics here, because nobody in the world knows how much eating charred meat contributes to the risk of developing cancer.
From a cook's point of view, you are doing it wrong. Well cooked meat is juicy and tasty. Overcooked meat is tough as shoe leather. You can continue doing it, if you want to, it is just irrational, like saying "I let my salad wilt for at least a week before eating it, I prefer it that way."