Aero bars on rental bike

I rented a bike for a race only once and that was for the San Francisco Escape from the Rock tri, no need for aero bars there .

From what I remember, Racine is pretty flat so aero bars might not be needed, but even if you want to put them on, it shouldn't mess up your shifters in any way.

Are you saying you want to put on your own cockpit (Base Bar with shifters/brakes, aero extensions etc.) onto a rental bike or simply put bolt-on aero bars onto the rental?  If it's the former-a decent bike shop could do that. Shouldn't hurt your shifters at all.  If it's the latter-YOU can do that yourself.  Just make sure the aero bars don't damage the tape/handlebar of the rental bike.

Before anything else, what kind of bike are you renting? A road bike, a tri bike? The answer would be very different...

For a road bike, you would need ITU (aka. Shortie) clip-on, which has nothing to do with the shifters at all... These types of aerobars will allow you to still ride the bike with a good balance (i.e. like a road bike, with the weight rearward).

For a tri bike, as long as the aerobars that's already on the bike are adjustable enough (and most are) to fit you, don't mess with it and change them out to your own... More than likely, you're not going to ride the bike a few weeks before the race, so cable stretch would be a major concern... you don't want to have a bike during the race that doesn't shift properly.