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.
Look on Gunbroker, you might find what you're looking for there, I've been hunting with handguns since I purchased my first hunting handgun in 1979, a T/C Contender in 41 Magnum with a 10" barrel. It wears a Crimson Trace 2.5 x 8 scope. Over the years it has had several different optics on it. I also have a Ruger Super Redhawk in 41 Magnum as well. It wears a Bushnell 2 x 6 Elite scope. I have 2 Nosler custom bolt action specialty pistols, one is in 6.5 Creedmoor, the other is in 308 Winchester. They are very accurate. What ever you choose practice, practice, then practice some more. Till you can put all 6 shots (if you choose a revolver) in an 8" circle at 100 yds.
I just got a CVA scout V2 in 44 rem mag today…going to try a Burris fast fire 3moa red dot on it. I have 550 into the weapon but may end up higher if I don't like the red dot.
I've taken a mule deer and whitetail deer with my Ruger Super Blackhawk 44 Rem mag with 10 1/2" bbl. It works well if you practice using the iron sights. ;o) It pretty much stays in the safe these days... This model pistol may be a cheaper option if you are worried about cost.
Like my OLD Contender, 30 Herrett w/4x Leupold & 357 Herrett w/open sights, both 10" bull barrells. Dozens of vermin and few white tails ventilated. Obviously, have to resize, cut & fire form cases but that's part of the fun.
Whatever cranks your tractor!!!!!!
I've got a Ruger Redhawk 44 mag that I inherited from my uncle that I intend to hunt with at some point. Put a Burris FF3 on it but haven't shot it yet.