If the output enable didn't exist you could move over to a variant of the 595 that has open-drain outputs such as the NPIC6C595-Q100 (which also have the advantage of driving higher voltages than 5 if you need it). You'd have to add PNP switching transistor and PWM on the high-side though with open-drain outputs. But the OE does exist!
Or you could directly PWM the outputs themselves (would have to re-update the shift-register chain at a very high rate of course).