Premium IV is a concierge service located on St Barth. One of the services that they offer is a VIP handling/expediting service for people transiting through SXM. They partner with a local SXM aviation handling service to do so. They can meet you at the jetway (if they still existed today) or the Transit Desk (if an analog exists today), fetch your checked luggage and transfer it to your SBH flight (negating the need for you to go through immigration, fetch your own bags, recheck them, and pass through the departure process), escort you to your gate, and potentially get you on an earlier flight.
The above is what is offered for people doing an in-transit connection at SXM. It’s worked well in the past. I don’t know if/how it will work today, given the jury-rigged status of arrival/departure at SXM. I also don’t know if/how they can assist passengers who are arriving at SXM, but departing from SFG.
As to whether or not you’ll need VIP help, nobody can say today. SXM needs to operate for a bit before anyone can say if it is working smoothly, or if it is a horror show. Next comes the question of what, if anything, PIV and their partner can do to expedite a passenger connecting through SFG.
Check the link below, and reach out to Vincent and Jim, the proprietors of PIV. Those of us who have been doing business with them for years, for other services in addition to the VIP handling, can’t say enough good things about them.
http://www.premiumiv.com/st-barts-arrival-assistance/st-barts-airport-service/
