Do you wrap your fingers around the bars and pull to compensate for the rocking motion while climbing? The pull uses back-muscles. I've found smoothing my spin and resting my fingers on top of the bars so I can't pull removes any back strain.

Decades ago when I started racing, I really pulled my back severely and was bed-ridden for a month. The cause was poor newby pedaling form and lots of yanking on the bars to steady the bike. Exacerbated by my really strong legs from 15-years of soccer. My back-muscles didn't stand a chance.

A chiropractor helped me stretch out the muscles and repair the torn tissue. Then it was PT-physiotherapy for recovery. Then gym workouts to strengthen the back muscles. The one exercise that helped the most was roman-chair back-lifts. I went from barely being able to go up halfway, to going up fully straight to fully up with 30-lb weight behind my head. That took about 2-months of training and I ended up doing 10-years of road-racing and 6-years of track with no back problems at all.