Shade tree, I think you can derust the cogs where they are with little trouble using EvapoRust...I've used it on all sorts of things including complicated assemblies and as advertised, it does not attack anything but rust. Clean the oil off of gears with blasts of spray brake cleaner, which can drip on out the bottom. Fill the thing up with Evaporust, the gallon you need would be about $20, rotate things as needed, give it a few hours. When you drain it out the ER will likely be contaminated by oil from bearings and stuff, but it'll be worth the cost over taking the car apart. Any residue can be flushed out with hot water, fill the thing back up with your gear oil and rotate all to oil the bearings and gears before re-rusting happens. This stuff is excellent and truly harmless.
Run car around the block, drain out some of the oil to clear out whatever remaining water based stuff leaches out of cluster and bearings.