The only way to book excursions on Coco Cay is through Royal Caribbean. It is their private island and they control everything.
The pros to booking your own tours is that they can be cheaper than those offered by the cruise line and you can customize a tour to suit your needs. The cons are that the ship won't know you are on a private tour and thus if something happens and your private tour is late returning to the pier you may miss the boat. The ships typically wait for late tours booked with the cruise line. The same is not true for privately booked tours.
For new cruisers who are not familiar with a particular port, I recommend they buy some peace of mind and go with the ship's tours. It really depends on the port. I've been to all the Caribbean ports enough to know my way around and what to do if a problem develops - so these days I normally arrange things on my own.....but not always. Sometimes a ship-offered tour can make sense.
Do you mean bike rentals for Nassau? Coco Cay doesn't have them if I recall correctly. No need for them since Coco Cay is so small....it is just a few acres of sand in the ocean.