Hey Jean, exciting project you have lined up there! My first impression is that 3kw is a BIG system for going completely off grid. That is 72kwh every day! Even if you wanted to drive 100 miles every single day, you would likely have 40 kwh left over just to run the house. I would suggest you start by doing an audit of how much power you think you are likely to need - and then design a system to meet that requirement. Unless budget is a non-issue, a big system like that is going to get pretty expensive pretty fast.
400 foot delivery distance is not going to be a huge hurdle, I have 500 feet of wire, but I am only moving like 100 watts. More power will call for higher voltage and thicker cables, but it should not be a problem at all.
My advice would be to start with the waterwheel idea. Turbines do gain you a little efficiency, but they are not without their own problems. They need a pretty sophisticated intake screen to not need constant cleaning (ask me how I know).
I think the project's credibility rests on a few factors: Budget, your own fabrication and mechanical ability, budget, availability of
local people to do the work if you are unable, local laws about mucking around in a stream (or distance to nearest regulatory body that would care about it), your tolerance for splashing around in cold water when something goes wrong, and budget.
You will not be likely to find a ready-made waterwheel for purchase, but there might be outfits that would build you one (this will likely be shockingly expensive). Waterwheel plans are a lot more available, I did a bunch of research and turned up some good PDFs about designs and such a while back, but cant remember now where I found them. I do recall that waterwheels have a very consistent efficiency across a wide range of flow rates. Basically anything between 20 and 100% of the design flow rate will yield 70-80% efficiency at the wheel shaft. This assumes you build a steel wheel with the correct geometry- a clunky wooden job will be considerably less efficient. You will also need to budget some efficiency losses in power transmission, the generator, the transmission wire and the rectifier if you use one. I think 50% of the shaft power is doable, but you might want to aim a little lower and not end up disappointed. Whatever number you come up with, shave off another 10% when you tell your wife to give yourself a little breathing room.
I seem to recall that there were rules of thumb about how many m^3/s of water were needed per meter of width, but I no longer remember what they were. The info is out there.
Power transmission could easily be handled by belts and chains. I would probably go with a chain from the wheel shaft, as it will not slip if it gets wet. Youd then need a second shaft that further increases RPMs, and drives something like a PMA with a standard v-belt. 3kw might be right on the edge of what you could transmit with a single v-belt, but I am not sure. The generator will have to be selected based on how you plan to transmit the power - wild 3 phase AC with a rectifier at the batteries probably makes the most sense, because even at 48 volts DC, you would be pushing 60+ amps at 3kw - and that would need a lot of copper to move 400 feet.
My site that i want to put a wheel on has even less drop, and so I am thinking about a breastshot wheel. Those sluices you have in place would make that style of a wheel a really easy approach. Ive even seen some that do not even use buckets, they just have curved vanes. The walls on both sides contain the water as it pours under the wheel. An image search will turn up some interesting examples. If it were me, I would be tempted to build 2 wheels, and use movable shutters to try and maximize power output at any given flow. Also, having redundancy is nice. If you are going to be doing the work yourself, you might want to start with a simple system and gain some
experience before trying to go for the maximum possible power.
At any rate, you have a great resource there, and I would love to see where you go with it.