I managed to get hold of Doron Swade who led the build of Difference Engine No. 2. He graciously replied to my random email about the Analytical Engine.
I won't reproduce the entire reply because it's long, and because the most important thing is something that he didn't say. He didn't say I was nuts! In fact, he too has been thinking about the best approach to building an Analytical Engine.
The conclusion that he had come to (as had I), is that the best way to approach building the Analytical Engine would be to begin with a physical simulation using a 3D graphics program with a physics engine so that the motion could be studied. My opinion is that this simulation would be a vehicle for raising the money to build the real device.
But even building the simulation is very complex because Babbage's plans for the Analytical Engine are incomplete and there are multiple versions. Getting to the simulation would itself be a research project to decide what would constitute an authentic Analytical Engine as Babbage would have conceived it.
So to build the engine there would be three major steps (the first two would be iterative):
1. Decide on the design of an Analytical Engine from Babbage's plans.
2. Build a computer simulation of the working engine to verify operation.
3. Build the physical machine.
At all stages money would be needed. First to pay for the research on the authentic machine, and second for the building of the simulation. Finally, money would be needed for the full build. Nevertheless, I think there's a significant community component as well: much of the simulation could be built by volunteers once the plans had been studied.
Estimating the cost will be difficult, but I can give a lower bound: the Difference Engine No. 2 build cost about £250,000 in the late 1980s. If inflation is to be believed that's around £390,000 today (which is about $620,000). The Analytical Engine would be bigger and more complex and hence more expensive and more research is needed.
So, I'm guessing (and this will need to be verified) that to complete a physical machine with historical accuracy would costs a small number of millions of £.