Problem found and solved. Two of the wires coming off the gear switch on the panel were shorting against a radio rack that had worn the insulation away from vibration over time. You can notice the small burnt spots on the wires in this photo http://www.imgivingnotice.com/wp-content/uploads/2015/01/gear-wires-Mooney.jpg
Apparently, the wires were moved away from contact when the A&P wiggled them on the jacks. That's why 20 gear retractions went fine. Starting the engine and taxiing created the short again. It took a little digging, probing, and moving things around behind the panel to find this, but my A&P persisted.
He replaced both wires from the switch to the cannon plug and all is good.
Makes me wonder about other wires in this 1984, 30 year old airplane!