whenever im doing long climbs or repeated stiff climbs i get similar back pain but nothing that lasts more than a couple mins after the climb. being on the nose of the saddle and wrenching on the hoods for hard efforts uphill for an extended period is pretty much guaranteed to piss something off on your body.

i would advise periodically switching up your hand positions (hoods, tops, bend, flats, rinse, repeat), your position fore/aft on the saddle, and standing/sitting. standing up and arching my back to stretch out while coasting usually helps if its particularly brutal or doesnt go away right after the climb