back pain difficult to diagnose when you see it never mind over internet. however, seems as though you have some mechanical back pain. where exactly is your burning feeling? groin, buttock, legs, feet etc. i would not recomend ab workouts but pilates or core work as these muscles more important for stabilising and controlling your back. you mention L5-S1, has anyone assessed you sacro-iliac joint - this can affect things. also seems you need to have myofascial release type massage, if this and anti-inflammatories are giving short term relief then stabilise with pilates. other considerations may be blood tests to rule out any inflammatory component to your symptoms, see and discuss with your GP.

the disc bulge will not go, the cortisone may help releive the inflammation and pain but if the disc bulge does reduce it will be over years.

also check your set-up on the bike and adjust as neccesary

you will need a sereis of exercises to do, but this depends on your exact presentation and their effect.

good luck