I'm not sure what you mean by "dump test".
On our side of the pond we do 'load testing'. That is where we apply an electrical load to the battery and then test it's lowest voltage. This also removes any surface charge.
Most times a good battery will load test just over 12 volts, but I've never seen one test at 12.65v. That sounds more like at rest voltage to me.
With the battery in the car, key in the run position, turn the heater fan on high and put the meter on the battery,,, what does it read now?
If it's lower than 12v, I'd say the battery is done for ... or if it has a removable top maybe you can add some water and limp it along for a few more months.
but everything does seems to point at the battery being the biggest problem. Second in line would be the cables themselves.