WELL.
We called a plumbing company we've used before and trust, and asked them to give us a second opinion. Turns out the problem is not a hole in the P-trap, but that the pipe has become disconnected from the shower drain. And he thinks there's no problem with the insulation, which is drying out, or with mold. Nor does he think there's an issue with the gas line.
So I may not have a disaster on my hands after all.