For under $500 you can get a CZ 453 which will shoot very, very well (groups in the .1x" range at 50 yards consistently) or you can get a less consistent but actually more accurate Savage MkII with a varmint barrel and a good stock (not the plastic one - Savage sells rotten synthetic stocks) and you can shoot groups UNDER .1" at 50 yards. Not every time of course but you'll average under .25" if you can do your part. At least mine will do that. You're right that you can get a fantastic shooting Savage for under $200 with the AccuTrigger. It will shoot about as accurate as the varmint barreled rifles only it will heat up and lose accuracy if you're shooting a lot of rounds. At any rate the AT can't hold a candle to the set trigger on the 453. It can be adjusted down to 8 oz. as it comes from the factory. It may be hard to find a 453 under $500 but it will be very close.
For under $1000 a Sako, an Anschutz, certain Remingtons (I keep forgetting the model # for the really good one) and of course the used Remingtons can be world class (like the 40X) as well as certain Winchesters. You'll find more 40X actions in BR shooting than you will about anything else. Also you can sometimes find a Suhl 150 for around $500. I saw 2 of them for under $500 just before Christmas 2 years ago and I've been kicking myself for not getting one ever since but I had a line on a 40X already built for BR shooting but I never got the money together quickly enough. I had already bought a .44 magnum for my bear problems and it took me a month to raise the money and by then the 40X was gone. That really broke my heart until I used the money to get a Savage 12 LRPV. That rifle is amazingly accurate.
If I were you I'd put about $350 into a Savage with a good stock and a varmint barrel and spend about $400 on a really good scope. You'll be shooting tacks all day long at 50 yards. Seriously. I can easily drive tacks at 50 yards with mine. I can do it with my 453 too. One of my best groups with the Savage had 4 really good shots and a flyer. The flyer made the group a .3" group which isn't that great for that rifle. But the other 4 shots measures a .01" group. Essentially that's 4 bullets into one hole that can barely be measured to be bigger than the size of the bullets. Once you get that close it's very hard to accurately measure the group size. But like I said it isn't as consistent as my CZ and it is very picky about ammo. The CZ will shoot ammo other than the stuff it likes best and still do well. The Savage won't. Both of them shoot their best with Wolf ammo BTW. That ammo is very good in a lot of rifles and it's a lot cheaper than the real top of the line stuff which in my experience doesn't shoot as well (like Tenex).
BTW here's the group I described where I put 4 shots in one hole out of 5 attempts. I used OnTarget
to measure the group. I can post an image showing the measurements if anyone is interested.