Hi
A suggestion
Arrive Nadi, go directly to Pacific Harbour via the Coral Sun bus - departs Nadi about 7am and 1pm daily. You can book a ticket on arrival at the office just outside the arrivals hall. FJ$20. You can also get a taxi that will set you back about FJ$150 (haggle) The trip s around 2 1/2 - 3 hours.
Stay at Uprising Resort. Uprising may have a shuttle service from Nadi airport. Best check with them.
Book your Shark dive as soon as possible as they get busy. Two operators. Aquatrek Beqa (my favorite), or Beqa Adventure Divers. Both good and offer very similar packages and are based at within 5 mins of each other. They dive different locations though.
Book a 2nd day's diving with your chosen operator to do the soft coral dives in Beqa Lagoon.
That leaves you 6 days.
Going to Taveuni or Kadavu will take out 2 of those days in travel and may limit your last day's diving as you have to take an internal flight. Having said that, the other posts are quite right that these 2 islands do offer some of the best diving in Fiji. If that was your choice, I would opt for Taveuni as the best sites are not as weather dependent as Kadavu's Astrolabe Reef is.
If you want to maximize your diving days, then you need to stay closer to Nadi and avoid internal flights.
That leaves you with Beqa area as you suggest. You could stay at Uprising or move over to Beqa Island itself.
In the former case, you would continue diving with Aquatrek or BAD. Beqa Lagoon Resort has their own dive boat.
However, my vote would be for moving to the other side of Viti Levu. Volivoli as I mentioned is a nice and very economical spot. Dorms are about FJ$25 per night and are definitely above average. Lots of locals hang out at the bar of an evening especially if there's a rugby match on TV. Getting there is a bit of a pain. Easiest method is returning to Nadi town on the morning Coral Sun bus, and then haggling a price with a taxi driver from Nadi to Volivoli. The airport taxis are much more expensive as a rule. Expect to pay about FJ$100 for the trip. You can do it by local buses but unless you are short of cash, it is not worth the hassle.
Volivoli is run by a Kiwi family, the Darlings. who are divers and good fun. They also offer private rooms in a shared house. If you want to dive with them but stay more upmarket, you can look at Wananavu resort which is a few minutes up the coast. Far more comfortable but not the same atmosphere.
There are also a few options to stay on the island just off shore, Nananu-i-ra. Same dive operation but a bit quieter.
The main reason for going to this side of Fiji is the soft corals and the chance to dive the Vatu-i-ra passage. Really class on a good day.
Return from Volivoli to Nadi is best by taxi again. It takes about 2 1/2 hours.
If you need to stay overnight before your flight, best bet is the Raffles Gateway opposite the airport.
I hope that gives you some more concrete ideas.
Cheers
Greg