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Not a good idea as you could easily burn out the charging circuit in your PE.
If you are looking for longer time of battery use then the best idea would be to get a rechargeable 12v battery pack that has a set of plug tips and use that plugged into the power socket.
If you do decide to add an extra internal battery you will need to find out which are the battery monitoring contacts and somehow devise a way that will enable the charging circuit to know the state of both batteries and so determine the charge rate - not an easy task.
Early in this forum there was a discussion about using different chargers and a couple of posts about using an external battery pack that might be worth your searching for them and having a read.
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