Presses can be very bad news, especially with a barbell; if the problems can't quickly be fixed by a number of adjustments including different grips, slightly different ROM, the use of dumbbells, cables, higher reps & shorter rests, it's best to avoid the problem exercises completely IMO. It's not gonna change, will only get worse.
Very little to do with muscle imbalances IMO, and everything to do with tremendous pressures on the joints that some people's bods will never be able to deal with. Those pains are warnings to stop what's causing it. I haven't done BB military or benches in decades. The substitute exercises are just as good or better, without causing problems.
Just switching to dumbbells and higher reps can be enough to avoid the pain, otherwise if it doesn't go away find substitutes. The substitutes can even be better than the original exercise.
If rest doesn't get rid of the pain, a cortisone shot as a last resort works wonders, but if the problem exercises causing the joint problems aren't removed the pain will return.