I've found that cleaning a lever action is not really any worse than a ML or even a single shot with a non-removable breech.
The way I do it is to leave a fired cartridge with a cleaning patch in it in the barrel with the action closed, I then use a mixture of rubbing alcohol and Murphy's oil soap and just keep running a fresh, wet patch thru from the muzzle until one comes out reasonably clean. I then remove the cartridge, open the action and run a couple of dry patches thru, finishing with a patch saturated with jojoba oil down. Then I use the cleaning mixture on a slightly damp patch and fish around the action getting the few bits of gunk out, then oil a wee bit with the jojoba.
Whether a ML or a cartridge rifle I never let a petroleum product near a black powder gun as black powder and oil don't mix, as the combination makes for a heck of a mess.
You might well consider giving the Holy Black a try as for me at least, its a lot more fun with all the smoke, the BP "boom" and the fact that it fits with the nostalgia etc. of the old guns.