I see no purpose in a .223 caliber as a "trainer" for 1000 yard shooting. The 22-250 you have would be better for that, it has the velocity to keep those tiny pills moving well enough to still have any hope of accuracy and/or consistency at that distance. I'm still in the Grendel camp if you are set on an AR for distance shooting. The BC and heavier bullet gives it the legs that the .223 is short on. As for a budget, count on at least a grand for a decent shooter, not counting the scope; add another $650 and you can get one that works for those distances. You do need a vise block for the upper .receiver, otherwise you risk tweaking the upper when you torque the barrel nut. As an example, here is the build list for my AR, and it's a guaranteed 1000+ yard gun, and it has yet to malfunction in any way since I built it; close to 750 rounds in the last year. All the ammo I shoot are my own loads, using AA brass and 123 AMAX and 123 SMK.
Odin Works 20" 1:8 button rifled, SS barrel, 6.5 Grendel, guaranteed 1 MOA accuracy (and it is)
Odin Works BCG
Odin Works adjustable gas block All that is $480 so far
Precision Firearms LMD brake $85
Mega forged A3 upper and lower receiver, $250
CMMG upper/lower parts kits, I forget the $$ but not much
VLTOR Gunfighter charging handle
Spike's rifle length gas tube
Geissele HSNM DMR 2-stage adjustable trigger $279
Geissele Mk8 M-LOK Rail $225
Magpul ACS carbine stock $90
Standard milspec buffer tube, spring and buffer
The scope I'm using is a Vortex Viper HS-T 6-24x50 $640
Warne 20 MOA scope mount, $130
Alexander Arms Grendel magazines x4 around $100
That comes to around $1500, and then add the optics and mount and you come to $2270. You can go cheaper on the trigger, rail, muzzle device, stock, and a few other parts, but you can't really cheap out on the barrel and BCG if you want a consistently accurate gun of any caliber