The diagnosis of SI joint disorder is probably wrong. Almost all SI joint pain originates from the discs at L4-S1 or nerve compression in the lumbar spine.
You note a “pinched nerve” at L3-4 but do not note what is causing the nerve compression or what side the compression is on.
Most likely, this is a typical case of radiculopathy from a compressed nerve. Your symptoms seem to match this disorder. You need to investigate further with a meticulous spine surgeon and I bet you will find the answer to this pain.
Donald Corenman, MD, DC is a highly-regarded spine surgeon, considered an expert in the area of neck and back pain. Trained as both a Medical Doctor and Doctor of Chiropractic, Dr. Corenman earned academic appointments as Clinical Assistant Professor and Assistant Professor of Orthopaedic Surgery at the University of Colorado Health Sciences Center, and his research on spine surgery and rehabilitation has resulted in the publication of multiple peer-reviewed articles and two books.