As Patsmom mentioned, you can use First Aid, but there are some disadvantages . One she didn't mention was needing to go back there to pick it up before leaving the park. More than one person has forgotten they left items at First Aid and had to go back - not an option if you leave at park closing.
I was going to mention Frio cases, but PatsMom beat me to it.
A Frio case will keep Byetta, insulin or other medication cool, although it won't be refrigerator temperature. They were originally designed for insulin, which needs similar conditions as Byetta.
According to the Byetta website, it needs to be kept at refrigerator temperature until first use, but once it is used, it needs only to be kept cooler than 77 degrees.
http://documents.byetta.com/Byetta_Medication_Guide.pdf
So, once you have started using a pen, a Frio pack or wallet will keep it just fine.
You can find Frio packs at various websites (including
Amazon). Here's one distributor that has a wide assortment:
http://www.frioinsulincoolingcase.com/
I don't know if they still do it, but at one time, Byetta provided free travel cases on request. Customers could request it from the Byetta website or 800 number. These were basically their version of a Frio wallet.
It looks like you can request one from this page:
https://byettaconsumer.myregistrationp.com/RM/registrationInputAction.action?ResponseSegmentId=6083