I recently picked up my most recent build in which I used the action from a Rem 700 SPS Tactical .223. Caliber is now a .22-250. I went with a Brux MTU 1-8" twist. OP, you have been given sound advice from the previous posters. Everyone has different ideas and preferences. I tend to favor MTU or Heavy Varmint. However, I am in the minority on this for both varminters and precision shooters. In your case I would suggest a med Palma or Sendero contour.
As for the Brux barrel...I am currently running Krieger 5R's (four rifles with Krieger), Bartlein 5R, Schneider P5, and now Brux. This Brux took all of 5 shots to see substantial reduction in copper fouling during barrel break in. My Brux barreled .22-250 is an absolute hammer for accuracy with 80gr AMAX and Berger VLD. I am as pleased with Brux as is am with my other barrels, which is to say I am very impressed.
As strange as it may sound, out of the barrels I currently own and shoot regularly. Only one brand uses button rifling instead of single point cut...and that is also my favorite barrel. It is a Schneider with Gary Schneider's version of polygon rifling or P5. It is on a .300 win mag with a McMillan G30 action. This barrel is super fast, extremely accurate, and does not seem to foul at all. My cleaning process includes using my borescope afterwards to ensure I cleaned adequately, etc. Every time I view this Schneider through my borescope, I am amazed at the mirror-like smooth finish with zero tool marks, etc. I don't know if all Schneider P5 barrels are as good as mine seems to be or not. Just saying I used to be adamant about only buying cut rifled barrels and then it had to be Krieger as well. This Brux is also beautiful through the borescope, so it is of little wonder that is produces excellent accuracy with nearly zero copper fouling.
Josh