For posterity, here’s the response from the local dive shop in Nanny Cay:

Thank you for your email. No the Rhone does not need to be dived on slack tide, so you do not need a tide table. However please be aware that the current on the Rhone can sometimes be extremely strong. We always drop a weighed line at the beginning of our dive to check on the current, if the line disappears behind the boat we go else where, if there is a slight current we advise divers, that it might increase during the dive, and if it does to stick close to the bottom, where the current is less. We have not had to pull off the Rhone for a while now!

We dove it on January 1st and had no current issues.