I'm not sure if this is a bug or actually intended, but ever since multiple fighting styles have been added, they haven't been added or been made available for the Blood Hunter. I'm aware Blood Hunter isn't official material, and as such, is probably not updated the same/as frequently as the other classes, but I felt I should make mention that only a couple fighting styles are available when there is now far more available. You can circumvent it by just not choosing a fighting style, and adding the feat that allows you to pick a fighting style from Fighter, as a temporary means, but it'd be nice to have the actual dropdown list updated!

This is the correct behaviour; Matt Mercer has not added these new fighting styles to the Blood Hunter. As such, they will not be available until he makes such an update.