"Coopertition" Words

[alopex_rex]

OK, I'm a bit of a language/linguistics nerd here, so you will have to bear with me, but:

We have "coopertition", which is a major element of this years game, for the first time since 2012, and of course one of FIRST's core values. It's an etymologically nonsensical combination of "cooperation", and "competition", but this is English, so we don't care about words making logical sense. But when we try to talk about coopertition, we run into a problem. We can say coopertition, but how do we say:

• To engage in coopertition (cf. cooperate, compete)
• One who engages in coopertition (cf cooperator, competitor)
• Pertaining to or conducive to coopertition (cf. cooperative, competitive)

When I try to form these words, I get things like "coopertate", "coopertator" and "coopertetitive", although much of the time my tongue gets tangled and I give up.

Surely I'm not the only one who's tried to use these words. What do you guys say?