Just thought I'd mention this, ANY firearm brands inexpensive rifles can shoot well or not, it's a gamble. You might get one that shoots 1/2 moa which would be rare, one that shoots around 1 moa which would be normal, or one that shoots like crap.
That's why those passionate about long range spend more money on higher quality gear, to ensure a higher degree of success when doing so.
It really is a good idea to save and buy an action platform that is better quality and a good foundation to build off of later on if you decide that's what you want to do.
^^^ but if spending more money is absolutely out of the question...
I like the RAP for what it is in stock form but I personally wouldn't be putting a bunch of money into it. At least it feeds and ejects well or at least the ones I've been around did. The loss is small if you buy one of these and go to resell later. I've seen them on sale for as low as $369. You'd only lose $50 or so on resale??? I've sighted in two for friends, they were 1 moa with cheap factory ammo and that sure surprised me. I owned on in 300BO, I just wanted to try the cartridge in a bolt rifle. After I messed with it for a while I sold it.
The HOWA I would put money into. It seems higher quality than the RAP and there are more aftermarket options.
- Forged flat-bottom receiver with integral recoil lug are the main reasons I like it, I'm considering getting one in 6.5 Grendel to play with, which BTW might be a consideration for you.
I haven't had good luck with Savage so I'll leave that well enough alone, poor quality, feeding problems, ejection problems, etc. Others have had good experiences, so there's that. It does offer different bolt heads to work with different cartridge cases, prefit barrels are available, and the floating bolt head is a very good idea.
Rem M700's have been good for me and they were "the" popular action to build off of for many years.
Tikka, although slightly out of your budget, has gained tons of traction in recent years. Likely the smoothest normal priced action there is and it seems very well made. Although, a friend bought two, one for himself and one for his son. Let's just say his son got the good one as far as accuracy is concerned - refer to my first paragraph.
Another thing, the old standard 7-08 is a very good long range cartridge. 30-06, 280Rem, 270, 25-06, 6.5x55, 260 Rem and other similar cartridges can get you by out to 1000Y. Put a brake on or buy a PAST recoil shield. Heck you might be able to borrow a rifle while saving.
What I did was spend extra money on a used rifle, at the time "back then" - I charged some of the money on the CC, and bought a used custom rifle. That rifle probably cost $3500 new, I got it for $1900. I used it 7 years and sold it for hardly a loss. That's a decision I'm glad I made! Then after the sale of it I bought some very high end gear.