Yours is an interesting and somewhat complicated question. Let's take it in steps:
Immigration: Your circumstances may vary not only from one island group to another but from one European country to another, and you will need to research what your visa requirements are for the island and Caribbean nations you plan to visit. Using Embassy websites is usually a good way to begin. When the information is not clear or you have questions about visa applications, I would consider using Skype to avoid phone costs. If advance visas are required, pay attention to two things: where can they be obtained and for how long a period is the visa valid after it has been issued. It may be of no help to learn in Puerto Rico that you must apply for a visa for the BVI's or St. Martin, but find the last chance you had to do so was in Santo Domingo, DR or Miami, Florida.
Imagine a typical Caribbean circuit: You may choose to stop at the Turks & Caicos, a UK dependency. You will perhaps stop in the Dominican Republic, an independent nation which will have its own immigration policy. At least you know where you stand with Puerto Rico, an American 'Commonwealth' (dependency). You will be drawn to the BVI's, which are again a dependency of the UK. Later, you will want to visit St. Martin - the French side does what France requires; the Dutch side is more independent of the mother country but influenced by her practices. Perhaps some other French islands will appeal, so back to France's requirements. There are probably some fairly consistent practices among the independent Caribbean island nations, but the home countries of the 'colonies' may differ.
Customs: Assuming you are not planning a lengthy stay in any one place, Customs' concern will be legal proof of ownership. If I were you, I would also want to carry a notarized Bill of Sale for the boat but that's more for your benefit than an official's needs. Customs officials see boats registered in one country and being captain'd or owner-sailed by citizens of other countries all the time. If your immigration papers properly identify YOU, and your boat papers confirm YOU as owner, then ownership should be clear to anyone.
They are mixed reports in the French islands about owners of USA state titled/registered boats being allowed to clear in. Sometimes this has been a big problem for the onwner; other times it is no problem at all. This sounds to me like an issue related to certain officials rather than an island 'policy'. Nevertheless, you should assume you MAY be turned away from e.g. Guadeloupe if you hit the wrong person.