Assuming an earlier introduction and/or spread of sugar beets, particularly in North America, would the value of Caribbean islands be reduced? For that matter, what effect would that kind of competition have on sugar-based economies?
There's several problem, while the process to extract sugar from beets was known early in the modern age, it wasn't developed for sevral reasons, one was the extraction process it is quite energy demanding, the second are breeding sugar beets took decades of selective breeding to develop to a useful level with government support, and even from that it took a century and the removal of slavery to become cheaper than cane sugar, the last are soil quality; sugar beet demand high quality soil to be grown efficient, which mean that other food plants can't be grown there, In Prussia as example which develop the sugar beet, the rich soil of Altmark and Magdeburg was use to grow the plant, but it only became possible with the adoption of alternative food plants which could grow in the sandy soil of Mittelmark, Uckermark and Neumark; the potato.
All this mean that we need several thing to introduce it early, early spread of the potato, strong unitarian state without access to oversea sugar production, and interest in state sponsoring of agricultural improvement and cheap access to coal. Honestly if you produce all those thing at once the effect on the Caribbian are going to be the least effect.