What size targets are you trying to hit at 1k?
We generally shoot at a 10" gong, which can be frustrating to hit when the wind is moving around, and can be frustrating to see when i plunk it right i front of a clump if grass.
My buddies hostage target (2-2.5moa) is fairly easy to see and hit.
Heavy rifles are generally easier to shoot, rifles with lighter/cleaner triggers are generally easier to shoot, rifles with stocks that are stable, fit you, and are comfortable are usually easier to shoot.
I can't think of a single sub500 dollar rifle that has all of those from the get go, but a number of 600-900 dollar rifles do.
Bergara HMR, and other long range variants of our popular actions. My dad shoots a Bergara Premier HMR which was 1300....would still leave you 7 for a nice scope.
Or go the other way and get a cheaper rifle and a nicer scope.
Under 500 id generally give up the stock to get a barreled action i can work with, and when funds allow add a stock.
I like the ruger Preds, Howa 1500s, Savage 110/10s, and Rem700s, all with heavier tubes.
An Axis would probably work fine, but need a bit more work.
The Patriots ive got are accurate, but a little too light of barrel for my liking.
You can sometimes get all of these on sale in a food long range configuration but usually your gonna be stuck with a base gun under 500.
Optics wise im pretty happy with all the optics ive used in the 350-550 range. My personal standouts are, Arken Ep5, Sightron Stacs..theres also cheaper optics ive been happy with like the Arken SH4 (lower glass quality), and USO TS12 (lower power, squishyish turrets, but great glass).
500-1000 bucks theres alot of good options, Meoptas, Athlon Midas Ares, Vortex Viper g2....others.
In that range, my personal preferences would be a Riton 5 Conquer or preferably a Sightron s3.
My Meopta HTR lists at over 1k as did my Conquer, similar models can now be bought for less than that tho.....
I shoot a 700 i put together from spare/used parts topped with a Riton 5 Conquer most of the time.