I have a T3 varmint .223. Also have a T3 light in 22-250. For calling coyotes I always grab the light. I'm not going to carry any varmint weight rifle anywhere.
I wouldn't go with a true varmint weight bbl myself, the predator models offer a good mix of bbl size to weight. Nothing sucks worse than packing a long heavy rifle around through brush. In your price range I'd look at the bergara b14 ridge. Or the Tc venture predator is in the $450 range and a heck of a rifle