pedantic answer: only way to melt carbon is to get it to over 3500degC 😉
The resin part is epoxy which is generally pretty much resistant to most things. Fume cabinet bases are often made of it if you need any evidence. There is a worry with painting some crash helmets but they were probably polyester resin.
You normally use solvent based 2k paints on carbon anyway. It should list the solvents on your data sheet or MSDS if you can find it and do a quick google for epoxy resistance but there is a slight worry about any thermoplastic tougheners which my be in the epoxy but I wouldnt worry too much about that unless you are soaking the frame in the solvent for days.
Or just use a different sizing. If you post the name I can take a look at the chemicals.