I'm not confident with Grackles, but I thought I'd get the discussion going.
From what I can gather, Great-tailed vs Carib is a very difficult ID. The main differences are bill structure, tail length, and overall size. Obviously, we cannot use tail length for this bird. It's also going to be hard to quantify the size difference without any scale reference in the photo.
That means we must rely on bill structure. The upper limit of bill size for Great-tailed Grackle is very large (
ML408512221 Great-tailed Grackle Macaulay Library). If your bird had a bill like this, it would certainly be a Great-tailed. Female Great-tailed Grackle can apparently have a smaller bill (
ML413743461 Great-tailed Grackle Macaulay Library) A direct comparison of the photos shows that your bird does not have nearly the same bill bulk as the male. The bill structure is quite similar to the female Great-tailed Grackle.
On the other hand, Carib Grackle tends to have a more short and conical bill structure (
ML579934521 Carib Grackle Macaulay Library). Direct comparison with your bird shows that its bill is longer than the typical Carib Grackle.
To complicate things, I found a photo of a Carib Grackle with an unusually long, downcurved bill structure (
ML349991791 Carib Grackle Macaulay Library). Maybe your bird's bill shape is within the natural variation for Carib Grackles?
In conclusion, I'm not sure. I hope someone else on this forum can provide better insights!
Also, for reference, here is an accepted record of Great-tailed Grackle in Couva-Tabaquite-Talparo from 2013:
ML91056111 Great-tailed Grackle Macaulay Library