Judging by your post, I have an idea of what you're looking for. However, I don't know what you already know about Fiji and what you expect.
Highland areas of Viti Levu don't have a good road system. The roads, if present, are likely to be unpaved. And, the combination of time of year plus locations (highlands) means weather is likely to work against you. In other words, rain plus dirt roads = difficulties.
In short, your idea of hiring a car is just not a realistic option. A 4WD is more likely, but even for that some road experience would be a big plus, if not a necessity. Other than that, local transport (sometimes a bus, sometimes a truck) will go as far as roads permit to certain villages, then it's walking/hiking/trekking.
The other important detail concerns local tradition. The settlements off the main highways are not the sorts of places a stranger just drops in on. Think: 'private property'. There's a bit of a ritual you'd be expected to follow before entering a settlement, and so on. Anyway, not the sorts of places a person would drive into, have lunch in a restaurant, then maybe try to find accommodation.
Having said all that, there are some good possibilities of homestay visits in upland areas, which you can organize through a travel agent. Yes, I sense you're looking for something non-touristy, but that's the reality of the situation. Considering the time you have, and time of year, the best bet is to visit the National Heritage Park outside of Lautoka/Nadi. The village/settlement of Abaca is the focus of this project. You could stay at a homestay there, take some walks in the area, possibly including a linking hike to another village nearby, also with homestay.
Maybe #1's link has some info in this (I didn't check). Otherwise, it is all easy enough to arrange after you arrive n Fiji.
It is probably possible to arrange a trip from Nadi/Lautoka to the more central highland area of Viti Levu, but there is more trekking involved here, and as I said the weather is more likely to be working against you. Better to wait until you're there and see what can realistically be done given weather circumstances at the time.