I just bore sighted my Romanian G AKM 7.62x39mm at 25 yards at home.
Made necessary windage adjustments of front sight to put it on at 25 yards.
Elevation seemed good.

Next I mounted my scope for bore sighting at same distance of 25 yards.
The AK has a side rail, and I am using Tantal Weaver style mount from Belarus.
The rings are Weaver 6 screw 30mm for my Bushnell Trophy 1-4 x 24mm scope.
Unfortunately the cross hairs were high and it took nearly all of the internal elevation adjustment to get it near 10X

Should I shim the rear ring so that I have some elevation adjustment left in the scope?
Will shimming just one ring be a problem?

I would buy some washers at Ace Hardware, and place them under the rear tang mount point on your rail.  I'm assuming that's what you mean by "shim the rear ring"?