Chosen Solution
I have the same problem with a1297. It’s not the battery . If you take off the bottom and heat with a hairdryer on the CPU & GPU the MacBook Pro starts fine. If its cold the problem comes back.
To see which processor needs to be re-balled. Heat just one or the other the GPU or CPU and see which one is the problem.
This looks like a defective capacitor. If you don’t have the schematics with the theorical voltage values, try warming them up one by one until you find the guilty one.