Have you patterned your existing full choke?
To me that would be key in determining what kind of a problem are you trying to solve, if any. Tighter is not always better at every range and with every gun.
I’m not a fan of 00 Buckshot (which is what most people refer to when they talk about buck) for coyotes. The pellet count is just too low and that lead to some unfortunate chases with coyotes who managed to find a hole in the pattern and only got hit in the foot or legs or rear.
#4 buck is very close to BB, but for me it’s more difficult to find now that manufacturers are cranking out coyote specific loads with BB. If you can find it, it’s a good option, though.
IMO it’s important to pattern test for two reasons. Obviously the pattern density is reason #1.
The not-so-obvious one is point of impact, or with a shotgun probably more accurately described as where the center of the pattern impacts relative to the point of aim. If you’re just using the bead as a sight, it can be rather surprising how far the center of your pattern is off from where you thought you were aiming.
I honestly believe that many people who believe they have a “pattern problem” actually have a point of impact problem where they are just plain missing the target with the center of the pattern. Both are equally important.
Grouse