I think you have a weak pilot and the updraft established by the hot fireplace basically blows it out after the main flame goes out.

I would start by pulling the pilot assembly out of the fireplace, then disassemble the pilot and clean with water and a little brush if you have something like that.

There's no reason to get the thermopile wet, just wipe it off.

There's a good chance the pilot orifice is partially blocked, if you have some 18 gauge stranded wire, you might be able to poke a single strand through the tiny hole. It's usually easier to just replace them. There should be a number on it, buy the same size.