^^^This might be the easiest solution--if you can find one.
Have had a T3x 1-8 .22-250 sporter for maybe 3 years now, which at the time came from a special run from Whittaker Guns of Owensboro, Kentucky. My wife has been using it at her pronghorn and deer rifle for a while now, and it works great with 70-grain Hornady GMXs--but also shoots great with lead-core varmint bullets in the 80-85 grain range.
Some have also mentioned Shaw barrels. I have had very good luck with their accuracy since Shaw (then E.R. Shaw) retooled around 20 years ago, and my latest is a 6mm-06 stainless barrel duplicating Remington sporter weight that I had installed on a 700 action in early 2020. Shaw installed it, lapping the lugs and squaring the face of the barrel, and it shoots several loads into 1/2" or less. They charge very affordable prices for their barrels and work--and offer 1-8 twist .224 barrels. (While they offer Remington-contour barrels in 3 different weights, they don't offer Mountain Rifle contours.)
Their standard available chamberings (no extra charge) include the .22-250, .22-250 AI and .22 Creedmoor. The last two are basically the same cartridge. As I pointed out in one of my early articles on the 6.5 Creedmoor that it's essentially the old RCBS wildcat, the 6.5/.250 Savage Improved. (That article was written so long ago that some people were worried about obtaining 6.5 Creedmoor brass if the round "failed commercially." I found it could be fireformed in one step with the Cream of Wheat method with .22-250 brass.)
I think you are off by 30 grains with the varmint bullets
Berger's twist-rate formula shows a stability of 1.44 with the 85.5 grain LR Hybrid at 4000' above sea level, when started at 3000 fps.
Have shot a bunch of 80-85 grain lead-cores with various 1-8 twist .224s in various chamberings from .223 up, in various temperatures. So far they have all done well. So which way is my assessment of "varmint bullets off by 30 grains?"
