I admire your solution to the problem .
Sometimes finding a fix that you can do now helps a LOT . I would put the shims under the bases for sure, by putting it IN the ring you changed the inside radius of the ring.
The cats meow way to do it is to find shims that will work work, then put a bore sighter in the muzzle, align the bore sighter with the rifle sighted in. Then take the scope OFF, optically center it...then adjust the shims and everything so the rifle ends up sighted in with the scope optically centered, or maybe at some other point that you like. For long range stuff I like them to sight in at 100 yards only 1/4 of the way up from lowest elevation, this gives me 3/4 of the vertical to use for ranging.
I have never had a bore sighter, I have done the above in my old shop where the bridgeport was by a window by clamping the rifle in the mill vise, then moving the vise until the rifle was sighted on an insulator on a transformer about 100 yards away. I had 6' tall windows but the glass was painted over on the bottom sash to deter the looky loos who love to see your lathes and mills and try to buy them "because your not using them" .