Alamo doesn't have transponders in their vehicles and I don't think any of the Orlando agencies do. They all use plate pass. Meaning when you drive through a toll and the system doesn't recognize a transponder, the toll gates take a photo of the license plate. Sun Pass then checks that plate against all of their registered vehicles, and then if there is no match, then they issue a violation.
When you rent a car and register your car on the Sun Pass website, you input the license place information and also indicate it is a rental car. This causes Sun Pass to register the toll to your account before going back to the car rental agency. There is absolutely no issue using a rental car at the tolls with your own transponder. We have done it plenty of times.
As for the OPs issue, just use the cash lines. Just don't take 417 as some of the exits may be SunPass only. If you want, go to Publix or someplace that sells the transponders or stickers after you get to Orlando, but plenty of people go to Orlando without transponders and have no issues. Just be careful of what exits you take. If you do end up getting a charge from Alamo it probably won't be as bad as you think, only $3.95 per day plus actual tolls. Might actually be a better deal than getting your own transponder if you won't be going back to Orlando any time soon.