DOM has been traditionally been connected to the rest of the world via ANU, BGI, SJU, and SXM. Pre-pandemic, there was ample service to all of those transit points. Keep in mind that immediately after Hurricane Maria in 2017, the airport in DOM was temporarily closed and that required the kind of overnight in PTP that you describe. However, once the airport reopened, that was no longer necessary and quite frankly would not make much sense. Unless you were traveling between DOM and France, PTP as a transit point is fairly useless. The connectivity between PTP and the US/LON/YYZ (the places most people traveling to/from DOM are coming from/going to) is minimal to non-existent.
With regard to DOM more generally, I am not sure who they expect to attract with the international airport. DOM will always be a niche tourism market at best. Ross University has moved. There is a VFR market but it's not particularly large. DOM has a population of just around 80K so outbound travel won't necessarily be high. I wish them well though.
Given growing trends of "health & wellness" tourism DOM is well poised to carve out its own niche. Increasing numbers of the more upscale market want something other than hordes of visitors in all inclusive resorts sunning on beaches. It will never be a major destination, but then that is its appeal.
Will its international airport work? I doubt it as even SVD and TAB struggle with theirs in terms of attracting the level of international airlift that makes these projects viable. Sandals in SVD might jump start better access, so it will not shock to see B6 joining VS in launching new service. DOM will never get Sandals though. Maybe the airport will get a less than daily AA from MIA and several private jets. DOM will use its passport $$ to subsidize some service.
Do not under estimate PTP though. France is an important source market for DOMs leisure sector., as are the French Antilles (though these arrive mainly by ferry). DOMs largest tourism markets are the Caribbean (meaning the French Antilles) and "other countries" (meaning continental Europe).
Depending on how BW handles its logistics at BGI it might be an easier in transit experiences than negotiating a transfer from AA/B6 at SJU to Seaborne. The appeal of this route will be if passengers can disembark one BW plane and immediately be directed to the intransit area to await the other flight.