Nope. Only thing that makes a difference is if you're upgrading a gold vs a bronze fighter, the gold is gonna take more coin and skillpoints.
Honestly I'd go for upgrading as you level since you're gonna need your characters to be beefed in order to get a high Prize Fight streak, which you need for just about anything as well as leveling.
It kinda just comes full circle if you upgrade while leveling rather than waiting, you'd be wasting a lot of time if not.