I was really struck by Dave Draper's most recent newsletter, in which he wrote that pressing movements are quite difficult/painful for him now, but he can still pull hard.
So you're right to be concerned about this imbalance. On push days, I'd cut back to two bench exercises per workout, picking different ones each time as per K-Narf's suggestion. I'm not even sure you need two, since you're also doing dips. Add one overhead pressing exercise as squeezer suggested. Substitute pullups for pulldowns, and consider moving them to pull day.
On pull days, I'd add low pulley cable rows. Deadlifts do help balance the horizontal pressing movements, particularly if you squeeze your scapula back at the top of each rep. So that gives you a pretty good balance of horizontal push and pull movements.
Personally, I actually do more pulling movements than pressing movements. Find what works best for you.