Depends on the nature of the Anglos who relocate to the islands. I could see, perhaps, an efflux from Alabama, Mississippi, Georgia, and Louisiana that might draw down the planter population of those states in the short term. That might trigger an influx to those states of otherwise-marginal slaveholders from the upper south; i.e., Tennessee, North Carolina, Virginia, Kentucky; all bent on getting larger plantations and the like. Now you have an opportunity for northerners to move into the upper south, particularly in developing towns like Louisville and Richmond, perhaps tipping the politics away from slavery. It might also trigger a migration of the last larger slaveholders from Maryland and Delaware, where slavery was eroding (particularly in the latter case), yielding abolition in both in the 1840s.
Now you have a situation where Missouri might still be a so-called border state, but Maryland, Delaware, and Kentucky will be Union states. Further, Tennessee and Virginia become the border states, with (say) Richmond as a Union outpost to thwart Confederate thrusts to the north since it stands athwart the key rail lines and roads. Virginia as a border state also gives the Union control over the entire Chesapeake, making blockades of southern ports easier.
I could also see some internal strife in Cuba and Puerto Rico, wherein the Hispanic elite would resent the influx of Protestant Anglo southerners. On the basis of the enemy of my enemy is my friend, perhaps many if not most would throw in with the Union, meaning the two islands would be battlegrounds.
This all adds up, IMO, to a Confederacy that is more volatile / harder to keep together than the one IOTL. Jefferson Davis--assuming he's elected president--would have his hands full just keeping things more or less together while trying to fight a war. And if the Hispanic elite get the upper hand on either island (not inconceivable since they were there first and know the territory), the island(s) in question are out of the war and out of the Confederacy. You might have a somewhat shorter war, ending before the 1864 elections, in which case Lincoln is virtually a shoo-in for a second term. And the shifts described above would seem to butterfly away John Wilkes Booth's actions.
Almost forgot: if the Hispanic elite does throw in with the Union, that may well diminish anti-Catholic bigotry in the north, as well as leading to somewhat grudging tolerance of Hispanic culture (notice I did not say "acceptance"). The Deep South, on the other hand, would remain as fiercely anti-Catholic if not more so. (In fact, in that climate, I could see the Orange Order gaining a real toehold such that the Orangemen and Klansmen might well become allies.) In the long run, you might find a handful of isolated Catholic islands in the South--New Orleans and environs--but you'd have to look long and hard for a Catholic parish in, say, Mississippi or Alabama.