Honestly, there is nothing to it. Antigua now has a brand-new terminal and everything works quite smoothly (except if you are running late, that means arriving after 4:00 pm, due to the sunset restriction in St. Barths). Be aware though, no matter which VIP company you will be using (there are only 2 of them), the agents are NOT allowed to retrieve your bags. In the old days, they had the red cap service but that's now gone. This means that agents will usually meet you in the arrival hall (downstairs at the bottom of the escalator), or upstairs as you come out of the jet bridge. They will give you a "fast-track" ticket which allows expediting the Customs clearance, but that's about it. Don't expect much more. You will still need to collect your bags yourself, clear Customs and check-in for your next flight (charter or scheduled, they all use the same check-in counters (# 1, 2, or 3). Once checked-in, you can proceed upstairs to clear Customs outbound and go through security to your gate. Again, there is nothing special about this transit, there are airports which are much more difficult than Antigua. Also, the walk is short. As you come out of Customs, the check-in area is just a few steps away on your right. So, you won't be dragging your bags much.
On-location VIP companies (all the others are simply outsourcing them):
Signature Flight Support ($70 USD per person, one way):
anumeetandgreet@signatureflight.com
Express Handlers ($46 USD per person, one way):
expresshandlers@gmail.com
By the way, airlines servicing St. Barths from Antigua also include St Barth Commuter and a few others (Anguilla Air Service, Trans Anguilla, West Indies Helicopters, Caribbean Helicopters, Fly Montserrat,..). Some of them will have to fly through their respective airspace (*) before going to St. Barths (for example the Anguilla-based or St. Maarten-based companies). All of them are using the above handling companies.
(*) As of April 2019, Winair is no longer authorized to fly DIRECT from Antigua to St. Barths. They now have to fly via SXM. This is due to traffic right limitations.
Note: Signature Flight Support also runs the FBO in Antigua (aka "runway 10"), located on the other side of the airport. The FBO is primarily used by private jets; no charter company going to SBH is using it (except if in transit between your private jet and a charter flight, just like in SXM).