Some thermometers come with a little stainless steel string of balls (like the pull chain for a fan). After the nylon string disintegrated on our new high-class floaty alligator thermometer, I just rigged it up to use the old pull-chain type leader from our old thermometer.
You could just stick it in the skimmer; it won't float out and probably won't interfere with the skimmer function.
I have a Taylor stainless thermometer that I took apart and installed an insulated, stranded 14ga. auto electric wire on about 20 years ago. Chlorine fades the color in the insulation but it is pretty much a permanent solution. Eventually the insulation will crack, but at pool temps it lasts much longer than it does in an engine, where it is good for at least 30 years. The exposure of copper to the water is infinitesimal (only at the end), and is contained within the thermometer.
I used a piece of an old telephone cord early this year due to the same issues of the old one turning to dust and it stayed all season long because A. it didn't break down at all and B. bugs can't climb onto it. I even left the length a bit long...like four or five feet so I could get the temp at the bottom rather than right at the top of the water.
I like the idea about making a line for the skimmer baskets...will have to try that one next year.