If you're not used to climbing and were always more of a spinner than masher, it's a predictable result of the change.

You have strong legs with good endurance. Unfortunately bicycling, especially spinning does little to strengthen the back. So now that you're climbing and there are stronger reaction forces trying to rock your hips, your back is working harder than it's used to to stabilize them. Unless you want to built a steel brace to carry rigidity from your shoulders to your hips, you're going to have to strengthen your back.

Odds are that if you continue to climb, but with some moderation and pain management, the necessary muscle support will develop naturally. You might get some relief by standing part time on long climbs, and maybe doing some exaggerated flexing between the climbs to relax these muscles. There might even be some side lift exercises that will help strengthen them faster, but I don't know of any offhand.

Keep in mind that these are basically static muscles that do isometrics against the back's twisting forces, so they can be prone to spasm and pain. Take it easy so things don't progress to where you can't ride at all, and give them a chance to catch up.

BTW- when my sciatica flares up from time to time, I've found that getting off and walking 100 yards or so makes a big difference and I can ride again. If I don't stop and walk, it reaches mind numbing pain levels.