The Mosin is no where near a big issue for the cleaning. But it does have too much kick for doing a lot of plinking. 20 rounds and my shoulder has had enough. Good fun though.
And then theres the simple fact that handgun ammo is cheaper to reload than the fancy rifle stuff. All of which suggests that you're on the right track for a plinking gun.
All in all the smaller and lighter the bullet the less costly they cost. So this suggests a lever or other action option that shoots .357Mag. It's noticably cheaper to reload these than .45Colt just for that reason. But we're only talking about 2 or 3 cents per round difference so you can decide.
For rifles the Marlin would be nice. I would not turn up your nose at a cowboy tuned Rossi Win '92 clone either. They are reliable enough, if a bit gritty feeling right out of the box. But when slicked up using the information easily found on the web they are sweet to work and shoot. They are so popular with the cowboy crowd because of the price that you should not have any trouble finding a smith that can do this work.
Another possible fun option is a Thompson Contender with the rifle stock and longer barrel. A break open single shot with a scope or nice target peep sight option can be a lot of fun too.
I know for myself the style of action and the need to chamber each round plays a big part in my plinking fun. In this regard the action is as important as the look of the rifle and the round it is shooting. It sounds like this is a factor for you as well. Sadly there's not a lot of variety in the common handgun calibers as one would like.
For example Chiappa out of Itally makes a Mini Sharps. It comes in a variety of chamberings but most are rifle or antique. But they do offer it in .45Colt.
http://chiappafirearms.com/product/829
Perhaps this would sway you to go that way for a plinking round and then you could add a lever action in that caliber at some point.
Found a couple more. Pedersoli makes a Baby Rolling Block in .45Colt. The rest are all in .45-70. Uberti also has a baby rolling block in .45Colt.
If you fancy these two as possible additions for later on you may want to start with .45Colt now unless you're already reloading for .38Spl and .357Mag for handguns.