Curious as to why you feel the need to build this rig? A 223 with a 52gr HP will kill coyotes out to 300 with ease and little need to turn turrets. Run a 40gr NBT and you can hold fur to 300.
Coyote hunting isn't a long-range game unless you want it to be, if I have a coyote beyond 300 yards I figure, I did something wrong. If I were looking to shoot long-range coyotes, I'd be looking at something bigger and better than an AR-15
That said any cartridge with more capacity than the 223 will move a bullet faster and flatter. The list is almost endless. The only limitation is how much money you want to put into it. A gunsmith can chamber a barrel blank for any cartridge and a die company will make you any custom die you want. If it doesn't work out, you can always stuff a different barrel into it
In WA, both east and west of the mountains one of my longtime favorite coyote cartridges was the 222 Rem for saving fur, and have killed coyotes in WA and across most of the west with 22 Mag, 22H, 218 Bee, 20 Practical, 222 Rem, 223 Rem, 22-204, 5.6x50R Mag, 22-250, 22 Sav HP, 6x45mm, 6mm-204, 243 Win and 6.5x58R Sauer. These were all intentionally called coyotes and all killed coyotes just fine..