Call Sally. She's a Dip MDT (Diplomate in Mechanical Diagnosis and Therapy, which is years of study beyond the regular PT degree). I worship at the feet of Dip MDT's.

She will want to look at your body first, and after she finds what is up with that and teaches you how to fix it, she may then want to look at you on the bike if the bike continues to re-derange your back. She'll look at you first, then the bike.

Honestly, I can't tell you anything about your back and your bike because I haven't seen you or worked with you. The best thing I can do for you is recommend you contact Sally or another Dip MDT near you. (Cert MDT's have also studied beyond the regular PT degree, but not as much as Dips)