I have done this with good results in Texas and on Bermuda greens. I did it in Chicago years ago and it seemed to work fine for bent greens. The USGA allows 10-14" of mix so if you have a constant 12" you can lower the ridges 2" and fill the lowest valleys 2" to reduce overall slope for today's green speeds, at least a bit. I wouldn't attempt to radically reshape the contours, just flatten them for green speed.
If you have deeper mix, then it tends to drain faster, and shallower mix tends to drain slower. Thus, if the deeper mix is now at the bottom of greens it helps drainage, and if the shallower mix is on the humps, it slows drainage and reduces dry spots, at least in theory.
Now, I am not sure if raising the high spots is as good an idea, but I think it could be done. You would probably have some dry spots that need attention. If low areas near the green swale exits are now shallower, you won't be cutting a pin in the last 10', but might want to make sure you have a "crow's foot" tile at the inside dish just to make sure that area doesn't hole water.
Just what is the slope in the areas that he wants to improve the drainage? I think 1.5% would be enough, although I would like more in cold climates.
I am also not sure if it would be political suicide or not. That would vary by club. But, in these times, I think if you presented the options as you describe and examples of where it has worked, and let them make the choice, it would be their call. In this day and age, presenting a reasonable, lower cost option has to be appreciated, I would think.
I should add the disclaimer that I haven't got a clue about where you are talking about, the on site conditions, etc. and always, and especially, I could be wrong. Certainly, on site advice would be far better than internet advice from someone who has never seen the real conditions.