Sorry to hear you are hurting and need to protect your back.
I would still be concerned that a really light rifle will potentially exacerbate accuracy issues. My experience with really light hunting rifles reminds me that I had to learn how to shoot them so I didn't cause excessive 'shooter induced variations'. Even then, I had more problems with the light rifles than I do with heavy benchrest rifles. For me, any pressure on a light rifle tends to accentuate variations in POI and I have yet to figure out how not to put some pressure on a rifle that recoils.
My 12 FV 6.5mm CM with the factory 26-in barrel mounted in an Oryx chassis with two mags, a NightForce 8-32x56mm Benchrest scope, in a padded Cabela's case weighs 17.1 lbs. Even without the Oryx chassis, the 12 FV in the factory stock was not something I would have wanted to carry around in the woods.
My Sinclair Gen 3 F-Class bipod, Protektor rear rest, range bag, and ammo all get carried separately from the car to the bench.
I don't have back problems but at my age, I'm not looking to create any, either.
Unless you carry a whole lot in your case with the rifle, the stock weight doesn't appear to be all that much different.
If you do carry lots of stuff with your rifle, you might want to separate some of it out to reduce the weight you have to lift to protect your back. Making two trips and carrying lighter is better than not being able to straighten up.
Some of my range buddies with back issues use a collapsible cart to drag their stuff around. Fortunately, I haven't had to resort to that yet, but it might be of help to you.
I wish you well with the new barrel.