We did exactly this with 10mo DS.
Just had private transfers from airport to Sofitel at Denarau, then went onto Shangri La at the Coral Coast the next day, also a private transfer (and the same return). It didn't add much at all to the cost of the trip and our travel agency arranged car seats (as well as portacots and high chairs for the whole trip).
From memory we had the choice of an early flight or a later one, we took the early one and getting the airport in time was painful but we were settled at the resort at Denarau in time for a restful afternoon by the pool, which was awesome.
We just fed DS from the restaurants, but he and I did both get a nasty case of upset tummy. I remember feeling terrible for not being more careful at the time but in hindsight it was probably the paw paw he ate, as even now 2 years later he would eat the stuff until it comes out his ears if I let him but just a few bites gives him an upset tummy.
I did take a bunch of those Rafferty's satchels but he wasn't having a bar of that when DH and I were gorging ourselves on delicious fresh food.
I was most cautious about the water, as our GP suggested we make sure he only drank bottled water. This was easy enough as there was Fiji water everywhere we looked
I have the most amazing photo of DS, asleep in the arms of a young male waiter at dinner, he snuggled up into the man's shoulder and just went out like a light. The staff at both resorts we visited were brilliant with him, he was a little celebrity and lapped it up.
We had a nanny at the Shangri La too, and went out for dinner by ourselves a few times. Our nanny was marvellous. We were embarassed by how poor the hourly rate was, and at the end of the week DH gave her a big tip and some extortionistically expensive choccies from the gift shop. I had done some research and taken with us some Aussie gossip mags even though I don't read them, as they are very pricey at the resort but it's a nice way to get your nanny on side
I can't think of a better place to take a child that age. Enjoy