I carry refrigerated medication when I travel and the ice pack is never questioned or even looked at. It is well over 3.4 oz and is often partially melted by the time I get there. This is explicitly allowed for medication.

The TSA site says ice packs for non-medical purposes must be frozen solid or they won't be allowed unless under 3.4 oz (which is unlikely.)

I also once had a frozen water bottle I was using to ice the medication, and they questioned it because it was starting to melt. This was in STT. I invited them to pitch it if it was a problem, and I also offered to open it and drink the liquid leaving only ice. They discussed it for a few seconds and let me keep it as it was. This took all of about 60 seconds and inconvenienced no one.

What's the risk? Worst case you lose your ice pack, and improvise something with plastic bags and ice cubes from a concession when you're through security. You're not going to get thrown in jail as long as you're nice about it.

There's nothing unreasonable about what you want to do. As long as you're willing to be flexible it'll probably be fine. And, if you are just trying to do this on the way down to the BVI, you may be fairly close to the airport and in a cold weather city, so appreciable melting before reaching TSA will be unlikely. Returning and taking a ferry ride in hot weather is another story - that's why my water bottle started to melt before I got to STT.

And no, I won't bake you a cake with a file in it if I'm wrong...