will I still take the +1 damage

Exhaustion? Do you mean pushing? Or is this a previewed Feat or BFC that I've missed?

According to the Marvel Danger Room Powers and Abilities Card, Power Cosmic team ability allows a character to not take pushing damage. If what you're talking about is "pushing damage" then they're safe. If it isn't specifically "pushing damage," they're going to take it.

It is a previewed BFC from Avengers. It modifies pushing damage by +1. So I'd say pseudosolider is right, you're not affected by this BFC if you have Power Cosmic.

This, to me, is like AP on zero damage

If you take none to begin with, you take none now (so Willpower is safe too)