(The rules don't recognize handedness, again.)
Here's the thing that struck me: what happens to the balance if we take the rule and make it one step more abstract? If you have a light weapon equipped, you can make a melee attack with it as a bonus action.
Okay. So, leave rapiers alone.
Dual wielding gets the slight buff I mentioned. Duelist is now giving you a choice of 1d8 or two attacks for 1d6 each.