mavic0406,
a closer look at the L4-5 cisc would be advantageous. Just speculating -- a herniated disc one where the annulus fibrosis (the rim of the disc) is breached and the acidic contents of the nucleus pulposis (the more liquid center of the disc) spill out when under pressure would explain pain over the broader area of the back. The spilling acid can travel down several levels of nerves triggering intense pain at each subsequent level. There is a test where they inject dye into the center of the disc until there is enough pressure to spill the contents. Under radiography the spread is revealed. But the proof is in the intense pain replicated by the test. I'm happy with your GP so many don't want to order MRI imaging. Now, though, you need to be seen in a pain clinic.
There is a minimally invasive procedure micro discectomy with annuloplasty to try and heal the breach in the annulus. I'd try a university pain clinic or an orthopedic spine clinic.
The test is called a discogram.
Best wishes