Double barrelled shotguns are luxury items for rich men.
If I had to have one under $2,000, I'd hunt a good SKB. I think the Weatherbys are from SKB, but probably cost more.
The Brownings made by Miroku are probably good guns. Not sure about spare parts or service for either brand.
In the UK, where doubles are the custom, Beretta seems to be the brand now most sought by those who cannot afford a London-made "Best" gun. I have a British correspondent who has a pair of Purdeys from her father, but who usually shoots game with a Beretta. Her usual quarry is pheasants and woodcock. She also rides to the hounds after foxes, which probably says something about her financial and social status!
However, the better Berettas considerably exceed $2,000. If you can find one of their simpler guns once sold here as the BL series, those may be under that sum. Jack O' Connor once mentioned giving one to his white hunter as a tip. He said it was a very nice gun for the money, but hardly equal to his favorite Berettas, which were in the SO series. It is worth noting that O'Connor also observed that SKB guns were designed by someone "who had had a good look" at Beretta's boxlock guns. That was Jack's wry phrasing. (Some Japanese- made rifles appear to owe Sako a design debt, too...)
Frankly, I don't think you should be looking at buying a double gun if you have to ask this question. They are not for poor men and they are not for men who don't know that market very well. They are also likely to be costly to repair, if you can even find a local gunsmith who'll do the work. Most so-called "gunsmiths" today aren't worthy of the name.
That said, I'd look for a SKB or Ithaca (made by SKB) that has seen little or no use. I wish you good fortune. A mystery author I know who owns a Browning BSS likes it a lot, but she isn't really a "gun person." Guns like hers were imported for only a short time, and I think were really made in S. Korea, maybe by a Miroku subsidiary. Her gun is either a true sidelock or has false sideplates. I think they may have been real sidelock guns. Browning can tell you more.
Start by finding and buying the second edition of the late Don Zutz's book, "The Double Shotgun." For most people, it's the best item of its type.