I built this year's ago from a $50 pawnshop Springfield model 120 single shot. I modified the stock so I could remove the action and barrel (held in with a single, knurled screw) & reverse them in the stock. I cut the barrel down to 16" & significantly reduced the stock. It packs down to around 20" & weighs a hair over 3lb. The sights are super primitive, but minute of skwerl head at 30yds.
It's rough as a badger's arse the days as its lived in my truck toolbox for about 10years, but it's eliminated a fair few small critters & one really unlucky little pig.
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Or get a barrel 16" long from Match grade machine they have some in stock usually. may need to have them cut one down to 16" or they have a custom order form:
Match Grade Machine (MGM) specializes in Thompson Center Contender barrels and accessories. TC Encore as well TC Contenders. Get custom built TC Contenders.
Take your Thompson Center rifle to the next level with Choate's all new telescoping stock. This stock is made of a tough polymer material for a lifetime...
The savage Rascal is cheap enough, single shot, with an ejector. You could do something like robtattoo has done above, or modify the stock to use a side folding stock with a lightweight wire buttstock. I have been thinking about doing the same thing.
I'll add one to the mix...Savage 42 take down available in 22lr or 22magnum with 410 barrel underneath...combo gun that's kinda homely but would fit your needs.