I used invisible thread on my Bernina Aurora 430 without a problem. The brand I used is YLI. Another option would be Superior Monopoly. Since it's polyester, it shouldn't "melt," even though I don't think that's the problem.
I would check the thread path and make sure the thread isn't hanging up somewhere - sometimes it can get caught on the spool. I had a spool of Sulky, and it seemed prone to tangling up, so a thread net might be useful.
Janet