If you can wait out the weather, you can usually find a window to make the passages between Turks & Caicos and the DR, and DR to Puerto Rico. The winds in June are starting to come out of the SE, and the biggest concern is Tropical Waves and Tropical Depressions which bring confused seas and gusty winds. Although early for hurricanes, keep an eye out for storms coming off the West coast of Africa, they're not unknown in June. Local weather reporting is often poor, so if you can find Wifi access to weather info, take advantage of it, especially in the Turks & Caicos.
Give a thought to going from Georgetown, Bahamas to Turks & Caicos, then Turks & Caicos to San Juan, Puerto Rico, bypassing the DR. You're then on the north coast of P.R., but if the winds are SE'ly rather than NE'ly, that won't be so bad.
Usually the most uncomfortable part of the crossing is the Anegada Passage, but we once crossed it in dead calm and had to motor all the way to St. Martin. That dead calm followed 10 days of really nasty weather when it wasn't even fun to take the dinghy ashore. Go figure.
How long do you plan to be in St. Kitts? As far as I know, the only reasonably safe hurricane hole is the lagoon in St. Martin, and I wouldn't consider it great in a Cat. 4 hurricane unless you could get into the mangroves or get hauled out and tied down on the hard.