Since I am in for at least another $125 and I now don't have a working tub / a gaping hole and burn mark in my bedroom, I tell him to do it right and just repipe. At this point, he retracts (maybe seeing that it would be a lot more work for only $50) and says that we should really just break the tile. I just want the guy out of my house, so I give him the green light on the $125 fix. This "$125 job" takes AT MOST 2 minutes of him tapping away with a screw driver. Then, he asks me to help him move the pipe over with my bare hands while he uses the blow torch on the other side. I end up burning my hands and he fixes it in another 10 minutes. So $125 for 12 minutes - awesome.
Sometimes a faucet leak can be stopped by replacing the rubber washer at the end of a faucet stem or cartridge. This is a more common repair on older fixtures when things were built to last and designed to be able to be maintained. Other times the whole stem or cartridge needs replacement or the bibb seat deeper back into the fixture needs replacement. That leads to matching up the right components and often times making a trip to the hardware store or plumbing supply house necessary due to the many varieties, makes and models of faucets. My advice is to make sure your Plumber has faucet repair kit on their truck before having them come out to assess the issue.