All right, very rough idea, but how about this:
Toussaint Louverture is never captured by Napoleon's forces, and remains in power post-Haitian revolution.
Dessalines never comes to power, and Toussaint doesn't order the mass-executions and the 1804 massacre never happens.
Instead, the new regime follows the spiritual footsteps of the French Revolution, and while dragging down the French colonist's absolute hold on power, they are either integrated, or expelled.
As such, Haiti isn't nearly as ostracized as it was originally, and actually gains some influence and power.
Using this as a template and inspired by the success, the 1812 Aponte Slave Rebellion in Cuba gains much more support from the free black population, and pulls off a similar victory.
We now have a free, black run Cuba and Haiti. French-Haitian relations are healed, and Cuban-French relations soon follow.
Spain tried to re-take Cuba, France intervenes, and the Dominican Republic is the next to fall to freed slaves, and the Unification of Hispaniola is a success.
Part of the cause for the US Civil War is (in addition to everything else), the South pressing to end relations with Cuba and Hispaniola.
During the US Civil War, part of the South's attention is focused on Cuba and Hispaniola; which sees the two come together for mutual protection, and gives birth to the Cuban-Hispaniolan Navy.
Following the end of the Civil War the two work to promote Emancipation Movements around the Caribbean, netting the islands you mentioned.
Around this time, the Communist Manifesto makes the rounds in the region, and the world's first quasi-Communist nation is born; more a confederation but still..
Friendly relations with the US and France (Britain is not so keen to have lost territory in the region), sees economic success.
Eventually, they get really cuddly with the Soviet Union in the 30's, and joint the allies with WWII.
However, with the end of the war, they go down the path of choosing the Soviets over the Free World; and so are very early adopters of Soviet technology, and recipients of Soviet aid.
That gives you plenty of lee-way and funding to build/do as you like.
Anyway, that's just off the top of my head.
Thoughts?