Stoegers are OK depending on their manufacture. They have varied quite a bit over the years. I would not buy any of the newer brazilian made models. But have seen an older model made in spain I believe that was a quality gun IMO.
Browning's will be out of reach in your price range, but that's not a huge problem. Brownings are nice but rarely found as a bargain due to their high popularity, especially models light enough in weight to meet my desires in a field gun. Look into Franchi's, Verona, SKB's, LC Smith (the new ones made by Marlin), Ruger Red Labels--all decent guns.
In any double--O/U or SxS--you don't have receiver length to deal with. Where that comes into play is that you can have longer barrels on a shorter gun than you would in a gun with a receiver like your BPS.
For dual clays/hunting use I'd look into 28" barrels. A 30" barrel might suit you too--and may be no longer than your BPS. 26" is getting a tad short for clays work but they make nice field guns if balanced well. Shorter than 26" I would avoid esp. if you want to mix clays in.
On the choke thing and steel, it's not quite as dire as mentioned for modern guns. Reason being that most factory steel loads pattern fairly tight as is, and every one places the pellets in a protective shot cup. Where you really run into problems is when you choke steel down in mod-full chokes. The shot cup doesn't help you if steel pellets get constricted in a tight choke--lead will deform and fly right through with no problems to the barrel--steel being harder is less likely to deform and more likely to cause barrel problems.
But you can get by with more open chokes for most everything when using steel. I don't know all the clay games but other than trap I think you can get away with using light mod or Improved cylinder chokes and they will also be as tight as you need for pheasant or duck work over decoys. If you get into long range pass shooting of ducks or geese I'd agree you have more of a dillema.