Z-sounds like you're on track, I'm a bit late to this party. But back in the day when I was a fiend for coyotes I had a similar discussion with a old govt trapper that I knew. He encouraged me to go Big 6, I did and I never regretted it.
Long and the short, I went 6/06 and it absolutely crushed yotes! Even on marginal hits that the dog would be whipering and bitching and moaning when hit with one of the .22's the dog would hit the turf darn quickly.
IMO, the seirous dog rifles begin with the Big 6's.
If, you decide you want to go for some pelts ( I did from the end of our big game season to the end of January) then I went to the 55 NBT. It blitzed along quite well, though it wasn't the best as ranges went long.
99% of our killing though was sub 500 yards. The longer shots were fun, farthest we went was 634 on a dog that just sat and barked at me
Our goal was to flatten 90% of the yotes we got into at 300 yards or less. We left the 10% for Mondays...
Best of luck to you!
(anything I can get to come to me via mouth call is the juice for me)