Wow, KBP, you've really limited your selections. None-the-less, I'll take a swing at making a couple of suggestions:

Walther Terrus (~$250)
Benjamin Trail Nitro Piston (~$270)
Hatsan 135 QE Vortex (~$300)

My Trail NP, however, is very picky about the ammo I feed it. So, be prepared to do some work if you get one.
I think the break-barrel design is not the best over the long term. As I see it, the main problem is that the rifle isn't solid because the barrel "breaks". For serious shooting, you'll want an optic of some sort and over time you'll find the barrel doesn't line up with the rest of the gun. Personally, I'd save up for a side- or under-lever. Then I'd save up and one day get myself a PCP rifle. The "Pre-Charged Pneumatic" (PCP) will require some way to fill its air reservoir. Do your research before making a purchase!
Otherwise, check out the Air Arms TX200 or the Walther LGU Varmint for a nice under-lever! Both are about $580-$600. But, they're VERY accurate without the need to carry equipment along to fill them when they get empty.
Enjoy!

Hey also check out Hatsan Torpedo

Hey, KBP
I don't know if you have bought anything yet, but I thought I'd let you know about a little rifle that I recently picked up. I wanted something inexpensive that I could carry in the vehicle with a synthetic stock that I wouldn't care about if it got scratched up. The ones that I decided to try were the Gamo swarm line. I liked them for the reasons I already mentioned and figured they'd be "good enough" for plinking and the random pest removal I might run across.
Two things I found that might interest you: 1) the Gen 2 10x magazine system on several of the guns was problematic. When it did work, it threw a lot of flyers. When it didn't work it would chew up the magazines so bad they were unusable. I think this is caused by the mechanism that rotates the magazine from horizontal to vertical and I ended up returning every one of them. 2) the original design ("Gen 1" 10x) swarm design has been flawless in the guns that I've owned. Yes, it worked so well that I bought a couple of them and gave them to family for their plinking/pesting.
Bonus: the old 10x style mechanism rifles really like the inexpensive Crosman Premier Hollow Point ammo!
~ Mondo

get a hw77k or the modern variant - I got mine in 1986 and its still awesome and groups 1" at 66yards on a rest

get a hw77k or the modern variant - I got mine in 1986 and its still awesome and groups 1" at 66yards on a rest

That a bit outside of KBP 's requested $200 price maximum, FXLover
https://www.airgunsofarizona.com/spring-piston/weihrauch-hw77k-air-rifle/