Moorea diving is easy, you will have no issues there. There used to be a Club Med in Moorea that took dozens of novice divers outside the reef every day. Good coral coverage, several shark species (white tips, black tips, the occasional lemon shark, and grey reef sharks out in the blue). Bora Bora is similarly easy, with the option of looking for the mantas inside the reef as well (which can involve quite a bit of swimming on a dive if they don't find them right away).
Fakarava and Rangiroa are going to be much more of a challenge. Either get your AOW before going, or risk the possibility that the dive guides might not let you dive outside the reef or within the passes. They advertise the requirement of being AOW, and I've seen them enforce this (yet, I've also seen them "relax" this restriction for a few divers). It's not just the current (in the passes), it's the depth to the edge of the wall in Fakarava, which in some places is ~25m+. Diving above this will mean you'll miss the full effect of the "wall of sharks..."
There are shallower dives in Fakarava, but to go all this way and not be able to dive the most famous site would be a travesty.
As an alternative, Tikehau has some excellent diving for all levels of divers, and will provide a taste of the Tuamotus (similar to Rangiroa and Fakarava). They also have a manta clearning station inside the lagoon, so easly to dive with the mantas here as well.
With 10 days for this trip (and the need to fly between most of the islands, which cuts into your diving/exploring time) I would consider limiting your trip to 2 islands, or at most 3.
I've aways said that Bora Bora is likely the prettiest island on the planet, so if you can afford staying there for even just a few days, it's worth it. Bora Bora is very expensive, in part because of the views and the experience. It is only outdone by the resort on Tetiaroa.
Moorea is worth a day or two as well, although, possibly not much more. There is a pretty serious algae bloom on the inner reef (due to the run off from the plantations on the island). Moorea's inner reef is not anything like it used to be, however the diving outside the reef is still reasonably good.
And then Tikehau - which will feel much less built up than either Bora Bora or Moorea, and still has some of that French Polynesian charm to it (as well as very good diving).
So here's how you know you have a successful travel plan for Fr. Polynesia:
If you can get away with 0 nights in Tahiti (or maximum of 1 night, either on the way in, or on the way out), you're doing well. Papeete is interesting for about an afternoon - the rest of the island will feel a bit like a tourist trap. The flight schedules are designed to have you arrive or depart at a time that makes you spend time on the main island. If you can arrive early in the day (vs. late at night) then you can get to an outer island on the same day that you arrive. If you have to spend the night, do so... just don't spend more than the 1 night needed. It's easy to get to Moorea from Papeete (either by ferry or flight, I've been on both - either is fine), or start in Bora Bora, transit back to Moorea, and then finish your trip in Tikehau. You can get back from Tikehau on the same day you depart from Papeete back to the US (if you plan your trip carefully).
I can also personally recommend Huahine, Raiatea, Taha'a, Tetiaroa, and Rurutu (if you can time it to be there when the humpbacks are there).
