There is NO problem shooting big shot in a choke tube that is very tight. IF the choke tube is rated for it, that all say on them the max size the type of shot and most have a speed rating too. The Hornadys stuff is VERY good but the 00 buck really limits the number of pellets. I reload and Prefer "F" lead. If you go to a heave shot there GREAT as well but pricy, On coyotes/turkeys you only shoot a few shots a day not like ducks, so price is not as bad as it sounds.
The old myth of running your gun with a tight choke is bunk, I use an extra full with BBB steel to, and slugs. IF you get the right choke for what your trying to shoot. ALL that said, even with the "Right" choke your gun may suck for pattern . You HAVE to try it to see, sorry, they ALL shoot a bit different .