Surprised the gent from Idaho hates the BL-22. I haven't yet met a Browning I hated. You should be able to use a bore snake to clean the barrel. Why you really need to get deep inside the action is a mystery to me short of submerging it in a lake.
Check out the action of any shooter that is shot a lot and never cleaned.
As far as Idaho goes, the most popular .22 is the 10/22 and the most lever actions I see are old 9422s or Marlin 39As...Lately the Henrys are selling like hotcakes.