I think you are going to have a tough time finding a shootable English gun for that price but if you ask a lot of questions, know what you want and are patient you may get lucky. The problem is that the market in the US is flooded with English guns that have not been maintained properly and/or have been shot to the point where they are well past their prime. Most guns you will find close to you price range will be out of proof (or close to it), or have been sleeved. That being said, last summer I found a HJ Hussey Sidelock Imperial for $2k. The gun has been sleeved (but had been re-proofed) and the stock was buggered but shootable with a slip on pad. That was a fantastic bargain. A coulpe of months ago I send the gun to a well known stocker who specializes in English guns and for $900 he did a butt-transplant and the gun fits me perfectly. I know that $2,900 is almost 2X your budget but for under $3k I have an extremely high quality London-made sidelock ejector that I can be very proud of. I got lucky but with patience and a good eye, you can find English guns that are good deals. Here are a couple of places to look.
Vintage Doubles
http://www.vintagedoubles.com
Hill Rod and Gun
http://www.hillrodandgun.com