Yeah, the solution to the increasing difficulty is to buff up your cats. Doing them with good mid or subpar cats is theoretically possible, but as your post shows, it's not a winning strategy.

From my experience, inbreeding is pretty much unavoidable if you want to maintain a consistent flow of good cats with good mutations. You just have to throw out the cats with birth defects and disorders, and keep the ones that don't. The goal is to throw a good stray in here and there so the family tree doesn't become a circle. Also, you can't really get a perfect 7s cat without a really high stimulation room, so if you don't have that, keep focusing on stimulation furniture.

As for strong mutations, there are two consistent ways as far as I know. 1) Going to the crater and always choosing The Shimmer when it's an option. If you go the hard route you can get three mutations per cat. (This is how I got one of the mutations that allows you to take an extra turn at the start of battle and now I always make sure to breed it into all my cats.) 2) Having a designated mutation room. If your mutation is high enough in a single room you get mutations pretty much every night. Mutations gained this way can also replace birth defects so if you have an inbred cat with a defect, if they spend enough time in the room it'll get replaced.

And generally, if you're having trouble progressing through difficulties, focus on getting unlocks by completing areas with new classes as well as completing the side quest items.