So do you have to get each character level 5 and level 7 before they can be a heavy or light?
I don't think we know for sure, and it could go either way.
It MAY require the character to level that high, because doing otherwise would unbalance things against new players when you're playing with and against strangers, but it might count as a customisation option.
I hope, for the sake of balance and fairness to new players, that you have to reach the required level to use them.