Hardsided cooler. We also used a strap, and yes, TSA opened and checked it, and replaced the strap. No Issue.

Coming home, we used the cooler for dirty laundry, and it was where we also tucked a couple of bottles of Pussers rum. Again, no issue.

Most of the charter companies will give you a few bucks for a used cooler if you don't want to bring it home.

Our cooler didn't have wheels, but we found it would sit nicely on top of my duffel which does have wheels. Easy. We wheeled it up to EIS from Trellis, as some of our crew were staying another couple of nights.