Finding the true center of a round plate should be easy. First, how did you make a circular plate without knowing the center? If you somehow managed that, just grip it around the outside in a self-centering chuck, spin it up, and use a center drill in the tailstock to drill the central hole. Further on in the process, after mounting the disk on a closely-fitted central axle, you turn the outside of the disk precisely true.
If no lathe is available, you can do it by taking multiple points around the perimeter of the circle and scribing arcs with dividers, all of the same radius. Keep adjusting the radius until they are all passing through a common point, which will be the precise center of the circle. Anyway, I'm sure you know all that, so at this point I haven't understood what the problem is.
FWIW, my usual method is to take a roughly-round oversize piece of plate, drill a hole roughly in the center, and bolt a closely-fitted piece of bar through the hole. Then I can mount the bar in a lathe chuck and turn the outside of the plate to the exact diameter required, true to the central bar. If you prefer you can mount the bar in a chuck mounted on the rotary table, and end-mill around the perimeter of the disk to bring it to the right diameter. Ideally you would then mill the gear teeth using a gear cutter of the same diametral pitch as the gear that has to mesh with it. Since I don't keep gear cutters, let alone a whole selection of them, I'd have to drill a series of equally-separated holes around the outside, just by indexing the rotary table. Then I'd use those holes as index marks to file equally-spaced triangular teeth around the outside. None of it is difficult but you end up with triangular teeth instead of involute ones, after doing even more work than cutting involute teeth with the correct cutter.
