I don't about that "don't load for expansion" thing, Art. & 50 gr .22-250 or a 60-75 gr .243 HP won't (usually) exit - pretty much complete bullet detonation within a coupla inches & well within the 'yotes body cavity.
Coupla ways to look at pelt hunting - either complete bullet destruction almost immediately (but within the body) or a "solid type" that won't expand & will do the "pencil hole" through both sides.
Coyotes are pretty small critters all-in-all & their vitals are measured in a 2-3 playing card--sized area (hearts & lungs). Their hearts can't be even as big as one card.
Whatever you can hit consistantly that size & that'll penetrate 12" or so will do 'em right there. Even a FMJ will do 'em plenty fast if a heart shot, but still expect a maybe 100 yard run till they expire. Most critters have an extreme will to live &
will run a ways .... (those "detonating" .22s & .243s seem to preculde this however ...
)
If trying a 12 ga, do yourself & the critter a favor & pattern the shotgun to see what kind of groups you get with various shot size, chokes & yards. Many are pretty surprised that their 9-pellet 00 buck load doesn't kill (maybe even hit) everything (anything) at 50 yards. I'd suggest a #4 buck as a minimum.