We are also heading down next month and using Dolphin Water Taxi for transport from STT to Tortola. I've been asking them all the testing questions etc. Here was the answer they gave me on 10/3 regarding proof of health insurance:

"No, you do not need "extra" insurance. just your own coverage for Intl travel. I would not worry about having to show anything from here on out. They just want you to have it."