An LP in the classical position is still going to be a problem, as then the main chords are too far up the neck to play for longer periods. Barre chords up near the head will still be hard. The fact you're also doing this with a strap to support the weight suggests you have not got positioning right at all.
I have no issues playing my LP sitting down in the regular non classical position. If you're supporting the neck when not using a strap, it's not sitting right to begin with, and that can be addressed.
There is no reason you should need to support the neck at all. I can only imagine that without a strap, you may be pushing the neck up to replicate getting the neck into the classical position and taking all the weight of the guitar into your fret hand.
The LP100 is a budget LP too. As LPs go, its very lightweight, being considerably thinner than Gibson, Epi Studio/Custom and Tokai versions.