Welcome to HH! I would first start my baseline a little higher ($450); as others have stated, it's gonna be tough finding a $300 hunting revolver. I prefer single actions, but that's just personal preference. I would suggest starting with a good (Ruger, S&W, or FA's) chambered in the finest cartridge ever designed (45 Colt).

It's been around over a hundred years and sees no end. You can load it from mild to wild and take most any game in NA. It's not as much factory ammo available, but that's just more reason to start reloading.

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It takes 43 muscles to frown and 17 to smile, but only 3 for proper trigger squeeze.