Low budget long range?

[Life's a Hillary]

Get a Tikka in 6.5M with the heaviest barrel possible
smooth action, accurate barrel

drop it into a KRG Bravo chassis
use AICS mags

spend more on a the gun now and don't regret it with a gun you can't upgrade. Bergara makes great barrels, but in the long run the Tikka will be the best bet

still definitely budget for long range

I'm picky on glass, so use what makes you happy but I'd recommend Seekins rings for mounting the optic

This combination can still be completed pretty cheap with a 10x or 12x SWFA and you would have one badass rig with plenty of upgrade potential if you want to drop some top tier glass on there.