Two tests with "anti" in the name have been discussed.

The Antibody test (mentioned by Mal) can be taken before your departure and is accepted for up to four months for entry into the USVI by water (taxi, ferry). The antibody test is not accepted for entry into the U.S. by air (which is driven by the CDC requirements).

The Antigen Test, also called Rapid Antigen Test, is now (as of 10 June) apparently accepted by the BVI for the test you do before departure (and enter the results on the portal). It's not clear yet whether the BVI intends to only allow some types of Rapid Antigen tests. Minister Malone's wording contained the word "approved" but there has been no clarification of which tests might be approved vs. not approved. This Rapid Antigen test is also accepted by the USVI for entry by water (taxi, ferry).

My reading of the update to the CDC Order is that a Rapid Antigen test would also (now), as of 7 May, be acceptable for entry to the U.S. by air but it must be supervised. If it is a DIY type of test, this means a video session to observe the process.

Results are usually available for both tests much more quickly than the RT-PCR tests.

Of course, all the rules can change at any moment so keep checking reliable sources.