Well, I've got two shotgun that could qualify as "inexpensive odd-ball"...
First, the Old stuff... A Marlin take down pump. It's been sitting in the project box, untill I decide to get involve in Cowboy action shooting!
Second, the "new odd-ball", A TOZ 21-12. I guess it's the russian version of the auto-5. I never fired an real Browning auto-5

But the design really does a good job at tamming recoil. And for a little over $300, they were a pretty good deal
Shown here with the top cover removed to show how the barrel moves in the action.