If you're new to springers, stay away from the magnum guns. Learning proper springer technique can be frustrating enough, without the extra recoil and general fussiness of a long stroke magnum. Regarding accuracy with a scope, stock design has far more impact than whether it's a fixed or break barrel. As far as accuracy in general is concerned, I've shot a lot of fixed and break barrels, and currently have 2 of each, I've seen no difference in accuracy based on how you cock it.
As far as which rifle, if you've been shooting PCP's, find a springer that has a similar design and feel to what you're used to. Just an example off the top of my head, if you've been shooting an AA S400 Xtra FAC, something like the Diana 460 will have a similar feel, I believe. It's worth noting that both the TX 200 and the HW 97 are purpose built for use with a scope, with a stock design that reflects that intended use. Both are very comfortable stocks, I had a 77 that I put in a 97 stock, and currently have a TX 200 Mk. III. Both are fairly heavy rifles, if you're intended use is tramping through the brush for a days hunting, they probably wouldn't be the best choice. The Diana rifles are a compromise design, but the lines are more classic, and work equally well with stocks or open sights. The Prosport has a wonderful reputation, although I've heard that some people have an issue with the cocking lever. Not familiar with it myself, so no details.
As to caliber, I feel that the medium powered springers are best in .177. Good velocity, flat trajectories, and lots of pellet choices. And I've taken game up to the size of large ground hogs with a 14 fpe .177, at ranges out to around 40 yards. 20 caliber has a huge following, but the pellet choice is extremely limited, as is the choice in rifles. .22's are best in PCP's, IMO. In medium springers, the velocity is rather low, and the trajectory gets pretty loopy. In magnum springers, this gets better, but they can be a real bear to shoot well or consistently.
As to an actual recommendation, with your budget, concentrate on the best. I currently have a TX 200 and an R9, both get top billing from me. The TX will most likely be good to go out of the box, the R9 will probably benefit from a tune, they can be buzzy or worse :-). Your budget has plenty of room for that, in fact, you could probably get both. I've had 4 Diana's, current one is a 34 which I like very much, enough that I sold off all my PCP's and switched back to springers. I had a 48 in .20, powerful rifle, but the stock design and the power level made mounting a scope a real bear. I also didn't like the side lever, it made cocking and loading safely really awkward. ALWAYS remember to hang on to the barrel or lever when loading springers! Also had a 350, I REALLY didn't like that one. Just too much work involved in shooting it well, it took all the fun out of shooting. Very unforgiving rifles, IMO.
Lots of options in your price range, these are just the ones I've had personal experience with. Hope it's of some help. Later.
Dave
PS Welcome to the forum!