Hi Rich, An old problem but a goodie. I signed up just to answer your Q.
Your problem sounds like a knackered relay. That it runs for a short time suggests corroded or worn contacts giving high resistance, overheating, which then in turn increases resistance to the point it won't let juice through to the motor. When the wiper is 'on' in intermittent setting the relay stays 'on' too and the motor receives a separate 'intermittent' signal. (Going by the wiring diagram).
I've just today solved my own E61's wiper issue and if I'd known about where the wiper relay is located I would have realised & fixed it when I did the hinge wires 2 years ago. In all the forum advice about wiper not working no one had said where the relay actually is and today I found it.
My E61 had the classic water damaged electrics in the wheel well and all the electronic modules were subsequently relocated to the left hand side of the boot by the last owner. Yes, the wiper relay (and others) was there too.
The BMW part number is 12631742690. Search that in RealOEM and you'll see where it is. It's relatively cheap at $25 on eBay. It's a fairly common part from the E46 forward (also an air injection pump relay)
At that price worth a crack to test.
Hope that helps you and others in the future who stumble here.