Put your voltmeter across the battery and tell us what the voltage is. It should be high 12s or low 13 volts. If the voltage is good then with the voltmeter in place, hit the starter button and see what the voltage drops to. If it drops a lot with that loud click, the battery is probably shot. How did the auto shop test the battery? Best way is with an AVR type instrument and a carbon pile.
Sure does remind me of a loose connection though. Will allow a small amount of current then when you draw a lot, the connections arcs and breaks.