In case it helps someone...
When I have a bag that I want to dunk the lining but not the whole bag, I pull it out and submerge it in a small plastic tub with the cleaner of choice, I prop the bag up on towels, phone books, etc.... so the liner is "hanging" in the water. If the water doesn't reach the edges, then I do that separately when the bath is over.
To dry the liners in this case and all the others that can be pulled out, I use a clothes hanger with pins and let it dry like that, so the liner is not touching the leather.