HI,
In the first instance, Iliopsoas (IP) syndrome comes to mind as this can produce a clicking or popping in the hip (most often seen in athletes who require a great deal of hip range of motion). It can be indicative of the iliopsoas tendon being inflamed itself, or the bursa underlying it. If in fact the bursa or tendon is highly irritated (as would be if the situation is ongoing) the muscle will want to shorten (contract) in the body's attempt to relax some tension on the muscle. This muscle (2 muscles actually - iliacus and psoas major) is a primary hip flexor - therefore shortened in the sitting position. they also act to flex the lumbar spine as the psoas part of the muscle attaches to the lower vertebrae of the lumbar spine.
An IP muscle in spasm or contracture would cause difficulty walking, compression of the lumbar spine, and can irritate the femoral nerve pathways from the lumbar spine to the thigh. Sitting would increase the compressive position of the lumbar spine and therefore aggravate any nerve irritation in this position which might not otherwise be seen in standing. I would also agree to you trying to get a lumberMRI in sitting or flexed position. and maybe (??) the sympathetic nervous system (responsible for those things like cold sweats, and potentially the pain responses that respond to crying???) runs along side your thoracic spine and might also be irritated by upper lumbar irritation (i don't know but it's a guestimation!!).
As for management (according to this hypothesis)- what are your hip flexors like? can you lay flat on your back with legs outstretched - do you feel a stretch across your thigh/hip?
are you painful to deep palpation in the lower abdomen (right down to the psoas level of muscle)?... a physio can have a decently uncomfortable prod to find out if your symptomatic side is worse or not.
I agree that it seems unlikely that McKenzie-esque exercises are going to be miraculous but the extension side of things will elicit a hip flexor stretch which might prove useful, in which case you may benefit from dedicated hip flexor stretching, and then some strengthening. I would also investigate whether in fact you do have an IP bursitis or tendinopathy and might benefit from a local corticosteroid injection. (MRI can also help with that diagnosis as well as a good physio assesment of your hip and lumbar spine).
And then there's the core stability, multifidi work and lumbar decompression work which i would recommend that you investigate the Sarah Keys methodology for.
Good luck, please let us know how you go.
MSK 101