My CAS lever rifle is a .357. Larger chamberings that you could share with a revolver include .41 Magnum (lever rifles in that caliber are very rare), .44 Special (the Uberti 1866 was offered in that caliber but not sure they are still in production), .44 Magnum and .45 Colt.
Whoops, forgot a couple of classic lever rifle chamberings (also available in revolvers): .38-40 (despite the name it's a .40 caliber and fits your criteria of a "larger" caliber) and .44-40.
A lever rifle is SASS-legal if it is chambered in a PISTOL caliber of .32 or greater. No .30-30 for example.
With cowboy shooters the .38-.357 is probably number one (lower ammo/component cost, low recoil); .45 Colt is probably the second most popular chambering.
I would take a close look at the .44 Magnum. You can load them down for cowboy matches. Some .45 Colt rifles have an issue with "blowby" that does not seem to affect the .44 Magnum as much. Better ballistics for hunting too.