I was looking at the eMed website and it recommends storage of the kits between 35.6-86 F. Is there any problem with transporting them in a carry on bag with the weather in St. Martin?

None at all. In fact you can't place them in checked baggage because of the temperature extremes in the cargo hold.

Taken from Abbott Labs - see below

11. Can the tests still work after sitting in cold temperatures?

Yes. A storage temperature range is very common in OTC health products. For BinaxNOW, it’s between 35.6 and 86° F and the test should be kept within this range. But if the test is stored outside the temperature range for a relatively short period of time – for a couple of hours up to a day or two – it will be fine to use, and it’s important is that test and its components be used at room temperature.