Alphadog has a point. I don't know how deep you are having to reach to get to that weight bolt head but a soldering iron might be your friend. It will take a while but thoroughly heating the whole weight bolt would be my first move if I have anything at all to grip.
If you don't have a way to drill a perfectly straight hole I would stay away from drilling a hole. I have taken out many a fastener broken off in metal. The customer usually breaks off a hard steel easyout in the hole before they bring it to me. Again, heat is the first move. Next move is a reverse rotation drill bit and drill. This loosens the connection between fastener and base material while providing some vibration. Sometimes this loosens things all by itself. If using an easyout remember that they are very hard, and usually very brittle. A crooked hole or not applying pressure evenly will often snap one.
An impact driver might work a little magic and they are cheap. Then I would look around for off the wall alternatives. The next size up star bit might hammer into that hole. What about the tools for removing the "one way" screws encountered on some jobs.
Unless you have a shop full of stuff and experience it struck me as I was typing, if there is a good cue smith around it might be best and cheapest to let him get the weight bolt out.
Haven't seen the material mentioned. Weight bolts are usually steel, stainless steel, or brass. Sometimes when somebody just wants to add a little weight right where they want it aluminum could be used. What that bolt is made of matters and might help to know.
My dad was magic removing damaged fasteners. I have to admit I never had his patience. I am much more of a get a hammer guy. If that doesn't work, get a bigger hammer. If a bigger hammer doesn't work, get a torch! Not the best advice working on a pool cue.(grin)
Hu