More training=more $$, always. So it would be up to whoever is paying for your classes, not the gym, so I wouldn't ask your coach if you could train with the team... I *would* ask your coach if they feel there is a better option for you (adding another class, going to a different rec level, etc) and then discuss those options with your parents.
At our gym, the only non-competing gymnasts allowed to train with the team are the developing (really pre-team) L3's that are new and/or not ready to compete (and they pay the same as the competing group, minus the actual comp fees).
If you are in rec and want more, you either have to sign up for more classes (in rec) or take the step to the other side of the gym "for real" and be part of the team.
Good luck!