What's your grip like on bench? Have you ever tinkered with it? Compared to your press, I'm sure you've got more in ya'.
In my experience I can tell if I need to dial back the pressing AW (dips) by whether I feel fatigued or not. I know that sounds obvious, but what I mean is that I usually can specifically feel if my triceps are tired, for example, whereas my legs might feel fresh but definitely not be, from mowing the lawn, or walking more than I realized the day before, etc.