20" eh?
Getting power uniformity can't difficult to get from even a perfect single mode gaussian beam profile. An ideal laser beam has power density related to proportional 1/r.. to some effect. The further out you move from the center of the beam, the lower the power density you experience.
Some multimode diode lasers may have an overall output which is more power- uniform, however such devices are difficult to focus. You could try starting with an underpowered 250mw module, and go straight into a beam expander, and maybe into another one after that until you get to about a 30" beam. Then by cropping out the outside 5" of the beam, you may get pretty good power uniformity by simply using the very middle of the beam- or in this case the inside 10" radius of the beam.
What do you mean to say about a first surface mirror?