I really and truly want to like this. It's the sort of 80s Cold War pulp thriller scenario one doesn't see anymore and that I have a bit of a soft spot for.
Also: Tropic Gull = Tropical? Ahh, good times.
If I'm understanding this correctly, you're trying to set up a sort of Cuban Missile Crisis 2.0 but seeing what happens if U.S. troops are actually sent in, which in turn leads to the kickoff of WWIII. For some reason it brings to mind one of WOPR's Global Thermonuclear War scenarios in
Wargames, but maybe that's just me. We see in your post a case of mistaken identity and a misunderstanding escalating into an engagement, but it's occurring at a time when the world isn't particularly on the brink and we need a valid reason as to why Cuba would want to escalate this any further. Stereotypical hot-blooded Latino requirements to avenge their honor only go so far here, and it certainly won't have the backing of the Soviet Union. They're already occupied dealing with Afghanistan, and at this point in the Cold War they're not about to come rushing in to save some client state's bacon when they mess up. They stood down for Libya and Grenada, and if Castro tries to keep raising the stakes here they'll almost certainly cut ties and leave him to his fate.
Now, if we want to press on with a full-blown shooting war between the U.S. and Cuba, there are a couple of options you could go for.
1) Arrange for Castro's death in some manner and put someone else in charge, something that leads to an internal rebellion, coups and counter coups a la Haiti that endangers commercial traffic in the Caribbean to the point of forcing a U.S. intervention to restore order.
2) Arrange for an incident (basically reuse your opening confrontation here) around the time of or post-Grenada like Matt Wiser posted where Castro is on edge and wondering if Cuba is next in the line of targets. If immediate fallout from the incident reaches the point where he believes that an operation against him is truly imminent, he could try to pre-empt with strikes to try and buy time. It'd take some serious handwaving in places, but it doesn't require Castro to go binonkers out of the blue.
Like I said, interesting premise, and a time frame that we don't see very often, but it just needs some more work. It's a lot better to put in a little extra research beforehand than asking us to play along with your new world rules. We're kind of sticklers for research on this board, but if you know what you're talking about we're much more willing to give you creative license as you spin your tale of destruction. Also, if you're dead set on this turning into a WWIII TL, don't be diabolical and sucker me in with expectations of a simple Caribbean war romp.