Ken gave you all the correct Info, even that he is since years away from Boats, his advise compared to today's experiences runs the same road, there vwas no significant change in the Rules/Laws.
to purchase a american flagged vessel and keep the US Registration you need to be a US Resident.
I purchased during the last year 2 sportfishing yachts in Puerto Rico, the same Code applies there.
You can as a Non Resident purchase the Boat there but You get the registration taken away, means no US/PR Insurer will insure Your Unregistered Vessel, and with such Vessel arriving in the DR You are obliged to run the Process of getting the Dominican Matricula/Dominican Flagged Boatpapers, because without them no DR Insurer would insure the Vessel neither.
to get that full Dominican Registration You have to Import the Vessel into the DR, doesn't matter how much or little You paid for the Boat, the DR Customs will run a inspection of the Vessel and Value it, and then you pay the import Taxes on that Value. once You have the Papers from Customs you can start the Matriculation Process.
as You do not want to use the Boat for commercial Use(here the DR Matricula/Flagg would be required), the best and completely Legal way would be that you find a way to register that Vessel under any Foreign Nationality, doesn't matter if that runs the flagg of Panama, the US(need to be s US Citizen), Lagos or any other Registering Country, in that case you can take any international Insurance for the Boat(make sure that the Greater Antilles and specifically the DR are included in the Area of Coverage) minus a Dominican Insurance, you bring the Boat over to the DR for recreational Purposes, request from the Marina de Guerra Headquarter a Permiso de Navegacion Recreacional, given for one year(and can then be renewed for the next year etc), and with that you can use your Boat here around the Island for recreational purposes when ever and where ever it pleases you, hassle free. it is what for example all the actually present Sportfishing Yachts do who visit every year the DR Eastshores for the Billfish and Tournament Season March - September.
Yes, You need to report your Ride every time you leave Port to the Port Authorities, but at some hot spots(where many foreign recreational vessels are present) like for example at Cap Can Marina, where I am based with my own Boats, the handling is very easy/hasslefree, if you leave just for the day with the intention to come back to your Port the same day without entering an other Port during the Day, then you just pick the Radio while riding out of the channel, name your boat number and the number of Passengers on Board(including the Crew Number) and go ahead, no paper to sign, no Authorities Office to be visited, no Authorities hopping on Board for a Check or such.
in case you plan to end your ride at a different port than your homeport you report that before leaving the Docks to the Office by Radio and within a 15 minutes the Officials will be present to do a quick check of the Boat(takes just a few minutes) and they fill in a Document for You which shows Your Departure Port and the Destination Port, that Paper You need to Present to the Authorities at the Destination Port, where they will check the Vessel upon arrival again. those procedures are of course only valid as long as you do not plan to leave the Country to visit an other Island, in that case the Authorities would also bring Immigrations Staff/Stamp Your Passport/get their Departure Tax per Person ect, but still no big Procedure, at least here at Cap Cana Marina.
the Keypoint for Your Case is to get Your Vessel Flagged under any foreign Country which accepts a registration from a Dutch Citizen, and get the Vessel Insured there.
just for recreational/non commercial Use I would not go the Import Street, it is long, it is very Rocky, and at the beginning you can never tell how long and how costy it finally will be to have it def done.
due bad experiences of a Friend with a Import Company with their Container from Europe I would myself never consider such company to Import real Value, and your Boat is def a real Piece of Value.
btw, for the greater Antilles the Cap Cana Marina is the only Hurricane Cat1 accepted Port by US Insurers, so it is a very safe Place to keep your vessel during the rougher Times of the Year, hence so many sportyachts from PR and the Virgin Islands come over here when Storm warnings are given out by the National Hurricane Center, as their own Homeports are only accepted by the Insurers til Tropical Storm Force as the Max. the advantage of having the homeport here is that you do not have big costs for Crew/Fuels etc for the short notice decided Travels to a safe Port, so over the Year the Cap Cana Marina comes out as quiet reasonable priced.
good luck which ever way you choose
Mike