The chip seems to only be a switch to switch on and off the coil when you draw, and in doing so also activate the blue LED, nothing fancy here.

Modifying this to be rechargeable wouldn't be to hard if you get this far, since there is a lot of space in the battery cradle, for making contacts and such, but you'd need to get a charging circuit into it somewhere which might pose a problem.

Wouldn't recommend it though since the coil has an expected end of life after certain cycles, and isn't replaceable.