You will get a lot of advice on this inquiry, but gunman synthesized the essence of it very well.
Hammerless Purdeys and H&Hs with 2.75" chambers, standard length steel barrels and "history" are not going to be found for under $15k - absent a sleeving or bad stocking job. You would be better off looking for "first tier" Birmingham or "second tier" London makers like Westley Richards, Evans, Powell, Wilkes, Ford, AA Brown, Boswell - they made high quality guns that may be found in great condition at or below your price range.
Guns are generally sold at auction for one of two reasons - they can't be sold elsewhere or they are collectible and the seller is trying to get a bunch of collectors to bid up the price. Even if you find a potentially good price without any lurking defect(s), the buyer's premium reduces the opportunity for a deal - a 20% buyer's premium on a $15k hammer price adds $3000 to the settlement price. And if you have someone prepare condition reports, the cost of those can add up if you end up kissing a lot of frogs first.
Many Brit sxs that could not sell in the UK have found their way to the States and are now sitting in dealers' inventories, overpriced and not in the original condition typically claimed. It may be worth considering looking in the UK and importing - the import fees are going to add $500-750 to the price but, for the right gun in your price range, that could be justified. You will also need to make sure you trust the dealer you are working with because returns to the UK can be a little tricky
