Pocketbrain

Yeah, something like that. I really don't have the schematics for the charging/protection circuit inside the Axim. It may not be charging the extender so much as just self-discharging with the external underpowered battery in place. Switching on/off of the charging circuit may be simply drawing enough current to discharge the internal battery.
Using a voltmeter, the charging input shows up as ~0V, so that tells me it should apear as open, so the power shouldn't be going into the external battery; it should have protection against that.

So, the discharge you see when the underpowered external pack is connected is probably the internal battery powering the charging circuit as it cycles. My best guess, according to the evidence at hand.

Note: I corrected overhead voltage in my response, it's actually about 1.6V. All the more reason to build a switching regulator, the power loss in the switching transistor would be a forward-biased transistor, less than a volt, and it would only be on part of the time, which would cut your overhead by a proportional degree.