While I cannot say you will find what your looking for withing the price range you specified, I can say that after 20+ years of loading my own for hunting use out of two 6" GP100s, I have settled on two or three bullets that will do anything I need getting done.
The Remington SJHP in either the 140 or 158gr, or the Speer Gold Dot in 158gr. Either of these has done the job very effectively on several feral hogs up into the 180 pound range. I haven't used them on deer, but would expect nothing less in performance. The few I have recovered are all in full mushroom and usually retain most of the original weight unless they smashed through a shoulder socket or similar bigger bone.
For a SD load I would offer up there are plenty of newer types and hypes, but for a long track record the Remington 125gr load is tough to beat. Those are some pretty impressive little pills.
I realize loading your own isn't for everyone, but for about what you will spend for a few hundred rounds you could pick up the components and begin your own loading and use what you want. As mentioned above even with a single stage press, you can load quality ammo and load it in the numbers you want. For a good while I was shooting upwards of 3-500 rounds a week from several revolvers and loading them back up within a couple of days on a single stage. If you DO decide to get into loading your own, look for bulk deals on bullets, and you will be ahead in the game. That was how I got to know how well the Remington's worked as a result they are the main bullets on my shelf, right along side my stash of Gold Dots.