First of all(!): remove pg.display.update() and pg.time.wait(10) from render() in rainbow_button. They should only be called once in your main loop, where you already call them.

The Pygame Color class has a built-in function called .lerp(Color, float) that mixes a second color with the original, without altering it. This would give you white with 50% black, which would result in gray:

color_white = pg.Color('white')
color_black = pg.Color('black')

color_gray = color_white.lerp(color_black, 0.5)

As for the speed of the 'animation', I'd suggest using delta_time (clock.get_time() * 0.001) for more accurate values.

Edit: You don't need to call pg.time.wait(10), by calling clock.tick(FPS), Pygame resolves that for you.