Hello,
I used to work on the Turks & Caicos Explorer II and still continue to live and scuba dive in Turks & Caicos (and work for Explorer Ventures, but in a land-based role). You should totally do a liveaboard in December and your daughter will be fine for her check-out dives! If she is older than 15 years old, there won't be any depth restrictions as the diving in TCI are quite deep often starting at 50-60ft on the liveaboards.
The water temperature is cooler in December, but not as cold as it will be in February. We always dive on the leeward side of the islands with the exception of French Cay which is out in the open. Since FC is not so protected, if the weather isn't right we don't head that way. The check-out dives would not be done out there.
I highly recommend diving on Nitrox the entire time you are here due to the deeper dives and the number of dives you will be doing on the liveaboard.
Feel free to ask any more specifics about TCI or our vessel - I'd be happy to help/advise!
Mia