別に is often prepended to a negative to soften it, like "not really".
Notionally, 別 (separate) expresses an explicitness or definiteness that is being negated.
別に来なくてもいいけど... "You don't really have to come, but..." (You don't have to make a separate effort to come)
別に悪いって言ってないよ... "I'm not really saying it's bad..." (I’m not making a separate statement that it is bad)
So here the friend is saying
"I don't really need it, I'll be taking one later anyway."
The 別 represents a strong, definite desire for a shower, which he/she does not have.
I think there should be a 。 after 別に in the sentence as written. (I see there is a space.) It’s a bit confusing without it. The 別に applies to an implied いらなない, not to the “この後…”