Any S&W or Ruger will take your whole budget just for the gun. Then you need a mount and scope.
You can get a Taurus Raging Hunter with a 6.75" barrel and a built in picatinny rail for just over $700. Then you just need to add a scope and rings. Or a red dot if you wanted to go that route.
The Taurus Raging Hunter is a comfortable and controllable 6-shot revolver. It features a 6.75-inch barrel, has a black finish, and has a fixed front sight and adjustable rear sight. This revolver also features an exposed hammer and an optics ready picatinny rail. Get this revolver and other...
I'll recommend something a little different. How about an HK45 or USP 45 shooting .45 Super? It has exceptional accuracy, ammunition is relatively inexpensive compared to the big bore revolver cartridges and the guns can be found used for relatively cheap.
I have a couple handguns that I hunt with. I have killed with a TC Encore with a 10in 7mm-08 barrel and a Smith model 29 in 44mag. I also have a super black hawk hunter in 44 that I haven't killed with yet.
In that price range you should be able to find a super black hawk or a Encore/contender
I got a model 44 Taurus 44 mag several years ago and have a Burris scope I bought off ebay. They stopped making this model and went to the Tracker or Raging bull. I heard and read bad things about Taurus, but this one shoots really good and after I changed trigger springs, it made a heck of a difference. I shoot the Hornady 240 gr and I'm inside a 2" circle at 100 yards with 3 shots.