I had a similar problem helping a friend with his 2011 GS350 AWD. The car was perfectly fine for months and then after sitting for a few days it wouldn't start. We tried jump starting the car and it wouldn't work. His cables were cheap, small gauge garbage. I used my 2 gauge cables and adjusted the connections on each vehicles battery until I got a good connection. After a couple of tries the car started up. I drove the car to the store and when I came back out the car was dead. Turns out the battery needed to be replaced. You stated that the battery in your car is new, but it may be defective. Take out the battery and have it tested. Also, is the battery to factory spec? Those "economy" batteries you see at the stores are junk. I believe the battery size is 24 or 24F depending on how the terminals are located. The factory spec is like 600 amps and 525 Cold Cranking Amps (CCA). You can buy a size 35 battery and put some cardboard underneath it to raise it up some. The size 35 battery is exactly the same dimensions as size 24, but is about a 1/4" less tall. I mention the different battery because size 35 batteries usually offer more power than size 24. It won't be a problem to change sizes either. My buddy's battery is size 35 with 700 CCA. Hope this helps.

 

 

Yeah,you can wiggle the harness.