Well, you've signed up for a bit of an adventure, but it's all possible, given enough time. Hopefully it's possible in a way that fits your schedule.
Liat can get you from EIS (Beef Island) to SXM (St Maarten Dutch Side).
Winair or St Barth Commuter can get you from SXM to SBH.
St Barth Commuter can get you from SFG (French Side) to SBH.
Voyager and Great Bay Ferries can get you from St Martin (French Side, Voyager, sometimes Marigot, sometimes Oyster Pond) or St Maarten (Dutch Side, Great Bay, Phillipsburg) to SBH.
Going from SBH to STT is a bit different.
You can go back to SXM using one of the options above, and Liat can get you to STT.
You can book a Shared Charter on Tradewind Aviation from SBH to STT.
Another option, if the cost is not a concern, is that St Barth Commuter will fly a private charter between EIS and SBH, and between SBH and STT. There may be other charter options too.
Here are some links which you may find useful:
http://www.liat.com/
http://www.fly-winair.sx/http://www.stbarthcommuter.com/en/index.php
http://www.voy12.com/en/
http://www.greatbayferry.com/
http://www.flytradewind.com/