Thank you! This particular antibiotic required refrigeration.
Here's the update for anyone who needs to know this information in the future.
Our flight to MCO was delayed 6+ hours, so when we finally arrived, we got a rental car and drove straight to DHS. Due to the flight delay, the cold pack wasn't very cold anymore, so I decided to leave the medicine at First Aid. It was a simple process: just fill out a form with some basic information and give them the medicine. I checked with the nurse to make sure they'd still be there after the 9:30 Fantasmic ended (park closed at 9:00). She said yes, they stay until after the last Fantasmic. As we were walking in the mass of people out of the park later that night, I looked at my phone and saw I had just missed a call from an Orlando number. They left a message saying the First Aid people were leaving at 10:15 and if I wasn't there in time, they'd leave it at Guest Relations. Well, at that moment, it was 10:15. I hustled to First Aid; they were already closed. I went to Guest Relations and as the CM went to check for me, the FA people came in to leave it. I was able to grab the medicine and go.
The next day, I took a syringe filled with the appropriate dose for midday. I was able to get a small cap for the syringe at a local pharmacy. This way, only one dose had to stay semi-cool and the rest of the bottle was able to stay in the hotel refrigerator.
Thanks for all the help!