How about - instead of tying OE to ground, connect it to an IO pin (any free Arduino pin) and a 10K pullup resistor.
The pullup resistor will keep OE high until you're ready for it, and when OE is high on a 74hc595 its output pins are in high impedance (disconnected).
When your startup & config code is done running in your Arduino sketch, do digitalWrite(yourPin, LOW) to enable the 74hc595's output pins, and away you go.
Good luck!