It has been a hellacious couple of days. Sorry I'm late on the reply.
To be brief, here is what I think is going on:
you have discogenic back pain, mediated by muscular control.
Basically one segment of your back is taking all the strain as others are slacking off, so one disc is smooshing out of shape more than the others. That one is complaining and referring pain out along the iliac crest (which is classic). The fact that you feel less pain when in better shape/practice indicates this isn't a defect, just a dysfunction. When your core is in better form, the strain is spread across all segments and you have less pain.
What causes the strain is a whole 'nother ball game, but I wouldn't give up the bike.
First I'd work on core: like CRAZY. (and yes, wearing the gear belt can really enhance the tendency of one segment to take all the strain.) Prone planks, side planks, and good posture at all times. I make my police patients do planks while wearing their gear belts. They love that...
If you can't get comfortable on the bike with huge core strengthening (like holding each plank position for a full minute 3 times in a row easily) then I'd look into leg length discrepency and bike fit and other funky things that really need a personal touch with a PT who can actually see your body and how it behaves.
If you disc is badly smooshed (like WindingRoad's) then bending your segments into extension can really help, whatever the cause. Lying backwards over a ball, bending backwards in standing, doing yoga "Cobra", all help smoosh discs back into normal shape.
Try core; try bending backwards before, during, and after riding; and try seeing a sports med PT (preferably one who has worked with cops and with cyclists).