First thing I would say is to listen to duhvoodooman's advice ahead of mine, especially when it comes to the function of parts in a pedal circuit. If he says the charge pump is the problem, it most likely is. I don't think you will damage the new part when you install it. I might hold my finger on it for a few seconds after connecting power, and disconnect immediately if it gets hot. (Note that you will not have to have the pedal in the "on" position for power to be flowing through the circuit.)
Your continuity test mostly makes sense to me. I don't think it's foolproof, but it's probably at least partly useful. There are better tests to do with your meter, however, and you've already done one of them (voltage readings on the ICs) that seems to have been very effective in tracking down your likely issue.
As for lead free solder, we
pretty much all hate it here. If you can legally buy the real stuff, I strongly encourage you to do so, and then throw the lead-free solder in the trash.