Try thoroughly re-cleaning the power contacts once again.
I had a similar situation with a 12 Vdc power connection recently. Wire harness continuity measured okay on my DMM and open-circuit voltages were present at all pins. However, whenever I tried to pump amps through the cable, the voltage would immediately disappear. Wiggling the cable would make the voltage re-appear intermittently.
Turns out the culprit was some tarnish on one of the brass pins of a brand new marine grade connector. Although I had inspected the connection repeatedly, the brass appeared to be bright and untarnished and so the connector remained low on the list of suspects. It wasn't until I removed one of the connectors from the cable and "re-installed" the connector directly onto the DMM probe that I was finally able to isolate the cause.
In my case, I successfully brightened up the brass male contacts using fine steel wool, but a pencil eraser or emery cloth should work just as well. If the problem is in the female contact, you may have to squirt some de-oxidizing chemical (e.g., DeOxIt) to clear up any latent corrosion.
Afterwards, I was explaining the problem to someone and pulled out another brand-new-in-the-box connector to show them. On this particular connector, three of the four pins were visibly tarnished, but the fourth pin was bright and shiny... just like the four pins of the connector I had actually installed. It's just bad luck that I randomly chose the "good" connector. Had I installed the other connector, the visible tarnish would've been a dead giveaway.
Dale