OK, I have a better idea of your needs now but I have to ask are you prepared to take the risk of burning out the charging circuit by adding the second battery?
That being said I think your best approach would be to get something like http://www.amazon.com/Rechargeable-3.../dp/B007RQW5WG
which is 12v only output only, or if you need 5v as well then there is http://www.amazon.com/3800mA-5600mAh.../dp/B0063EYY5Y
or even http://www.ebay.com/itm/Lawmate-Long...-/140656640010
I may sound as if I am preaching to the choir but don't get so stuck on the idea of adding an extra battery to the existing one in your PE and not look at the easier alternatives because if you do there is a very good chance the next problem you will have is how to repair the charging circuitry.
What ever you decide it has to be your decision so, good luck.