Originally Posted by w1ksz
The Green LED next to the charging connector is flashing when the charger is plugged in and I can run the Edge with the charger plugged in.
I still think it's the battery.
From the symptoms you describe you may have a faulty battery and unfortunately, unless you can find a unit that is damaged but having a good battery or you are prepared to buy a thousand of them, there is no hope of getting a new one (at the moment there is no way I can justify ordering such a large number of batteries just to have 99.9% sitting on the shelf).
On the other hand you may find, when you open your PE, that the battery has a loose connection or there may be a dry joint at one of the components between the charging circuit and battery.