You would rent the car and drive North along the Atlantic (Eastern) Coast to Speyside and Charlottesville. That's where the crazy currents are. In that area, look to Manta or Blue Water Inn, or just do some research and hook-up while you drive into Speyside. This is where you can see the "world's largest brain coral", the outlying barrier rocks, and appropriately named dives such as Heart Attack, Washing Machine, and African Express (Just consider that name for a moment, the meaning will likely come to you). Certainly no reputable charter will take you to any of those 3 non your first day of dive charter.
The South end (Pigeon Point and Crown Pt), you're pretty much-so going to get dive ops that cater to the Cruise Ship crowd no matter what anyone might say. It is what it is, and everybody gets treated the same, at least for the first day of diving. It's a universal in the world. It is the more developed end by far, and with that comes all of the access to good and bad.
Again, if you have bunches of money and want to stay at the Southern entry point, consider the Hilton and their resident dive operation- it is just up North a bit on the Atlantic side- kind of across the island from Pigeon Point, but still maybe only 20 minutes by car away from each other.
Some South end ops tout that they have access to North End sites, but once you look at the map and calculate what they have to up-charge you for fuel... plus the pounding you're going to take getting there- drive up to Speyside instead.
Tobago is one of the very few Caribbean islands that I would be okay with driving- the roads are very good, although horses can appear out of nowhere, the roads are twisty and mountainous, no reflectors, and the typical crappy rental cars with bad headlights and useless wipers (another island constant). It is a great island to explore along the North End and come down the Caribbean coastline. Waterfalls and such- some you can dive under~ but not on a two day stopover that includes diving.