Bullseye is generally considered a little too fast-burning for .357 Magnums. And large pistol primers will not easily fit into .357 cartridges!
A fairly large charge of fast burning powder would probably not be the sole cause of bullet instability. I would first look into bullet fit, hardness, lubrication and condition of the cylinder throats and barrel. Those variables can add up to result in key-hole shaped bullet holes. I would be interested to hear what the paper targets look like from 15 or 25 yards. (Any key holes? What's the groups size? Chrono data can be helpful, too.)
You certainly don't need magnum primers to ignite Bullseye unless it's 30 below!
p.s. My old Lyman reloading manual lists 4.5 grains of Bullseye as maximum with a 158 grain lead bullet.