Suggest Treatment For Persistent Lower Back Pain
I read your question carefully and I understand your concern.
Given the history with shooting pain with abnormal sensations reaching down to the lower limb, the most likely cause should be a nerve root compression due to a disc bulge or arthritis bony spurs. Initial management is conservative and consists in over the counter pain killers like Ibuprofen and physical therapy.
However if the symptoms persist for over 4-6 weeks like in your case then imaging is advised in order to confirm the diagnosis and evaluate the degree of compression as well as the need for a more aggressive approach such as local steroid injections or even surgery.
As for who is the most suited specialist I would say that apart from the family doctor the initial recommended specialist would be a neurologist. Afterwards depending on physical exam findings and imaging results there may be need for consultations with a neurosurgeon or a physiatrist, but the initial step should be to consult a neurologist.
I remain at your disposal for further questions.