DR is probably one of the better prepared for the Cuban competition. The DR isn't as dependent on the US market as some nearby islands (Puerto Rico, Bahamas, Jamaica; Cancun -not an island, but a major player in Caribbean tourism-, Aruba, and most of the other islands all depend on American tourists for well over 80% of their visitors, in many cases well over 90%). The has been competing with Cuba for the Canadian/Latin American/European markets and it has done very well.
The influx of American tourists might decrease for the DR (and across the Caribbean) as they head for Cuba instead, but they will displace many Canadian/Latin American/European tourists that vacation in Cuba. This displaced group is used to mainly European AI chains (the European AI chains in Cuba are the same players in the DR), and the DR is the best prepared to accommodate that group. DR has the hotel chains they know and trust, the flights connections to their airports (Punta Cana has the most flights to Europe of any Latin American airport and most major Canadian cities have direct flights too), the trained hotel staff that speaks their languages and/or knows the treatment they are used to, etc.
If we take Puerto Rico as an example. They overwhelmingly depend on American from the eastern seaboard. They hardly have flights to Europe, hotel chains are overwhelmingly American and mostly non-AI. Puerto Rican staff in most of their hotels probably understand English and that's it as far as non-native languages. There is no way PR can get ready in time for the Cuban competition, because once Americans start to head more towards Cuban than PR, they are going to have a hard time attracting the hoards of Europeans/Canadians/Latin Americans displaced from Cuba. Not many Russians, Germans, Italians, French are going to visit PR if there are no or too few direct flights when just across the Mona Passage thousands of Europeans from all over Europe arrive every single day in Punta Cana.
Jamaica faces a similar dilemma as PR, but at least Jamaica has the AI resorts even if only a few of the European chains. But they also face severe restrictions due to sparse flights to Europe/Latin America and their hotel workforce is overwhelmingly trained to cater to American quirks and nonsense which doesn't always rubs well with Europeans. Another negative for Jamaica is that its not a Latin country while DR is probably the closest thing to a Cuba without being in Cuba. While many people don't care where their resort is, a lot of people that travel to Cuba do so also because it is a Latin/Hispanic place with the vibe to go along with that. People that like to listen to Spanish/Latin American music by the pool might not get in their vacation mood listening to reggae or hip hop or whatever is not Latin.
Point is that DR is much better prepared than most of the Caribbean.
Cuba also needs to Americanize its hotel offerings. Americans are very picky and whiny people. Europeans take it as part of the experience when they have to eat in open air restaurants with the breeze soothing their skins, etc. That same scene is a nightmare for Americans, because most will not feel comfortable if there's no A/C, and the they will complain about the odd bug that flies through, get annoyed at the slower pace of things, and a very long whiny and annoying etc.