The only lever shotguns that I am aware of are the Marlin and Winchester 1894's (9410's) which are no longer manufactured, and the Winchester 1887, both originals and the Norinco's.
The 9410's were overpriced and would only take 2 1/2 inch shells. A nice piece, but not overly functional. Probably why they were discontinued.
I am not aware of 1887's in 20 ga, and I can tell you from experience that there is a reason why the Norc copies aren't imported anymore - without fine tuning and modifications, they are nothing more than a novelty with a cylinder bore. Slow for cowboy action, and useless for hunting. Also there is a safety factor in that they will fire out of battery if you touch the trigger while racking the lever - it really bites the fingers when that lever flies open.
Honestly, I don't think that there is anything out there in a 20 ga lever, or any lever shotgun that will fit your needs.