It looks pretty easy to "add a vehicle" on the sunpass site. The only reason you would need to add a rental would be if your sunpass isn't read by the toll correctly. You would be consider a violation, and a photo taken of your plate. Since the plate is registered to a car rental company, they would in turn track you down and get the fine or fee from you.
If you register your rental car as another vehicle, Sunpass would see that your transponder is for that vehicle and not involve the rental company.
The website has start and end times so that you can register a rental for just the period that it is in your possession.