SDA Serial Data
SCL - Serial Clock
NTC - Temp thermocouple.
Here is a good read. It basically a SMBus battery
http://batteryuniversity.com/learn/article/inner_workings_of_a_smart_battery
Most of the aftermarket batteries are 2 wire or 3 wire, which makes finding a good candidate more difficult.
We need a picture of the charging circuit chip and who makes it, then grab a data sheet. Most are for single pack charging so paralleling with load balancing seems unlike to be built in to the device. It's possible but not something I will probably be content in letting my child have on their lap. That leaves a larger capacity. Here is the manufacture site
http://www.tclbattery.com/read_products.php?smname=%D2%C6%B6%AF%B5%E7%D4%B4%CF%B5%C1%D0
The Nabi battery is in the mobile power section, you will have to switch to Tablet and see if anything there. I tried search for the higher capacities and didn't come up with much in limited searching that can be bought anywhere.
5.1mm x 65.5mm x 111.5mm, plenty of room in Nabi but that gives a general idea. I imagine if you can find a 5 wire battery of higher capacity it will work great.
Earlier I saw a question about jumpering USB to wall changer plug. Hard to tell from the pics, they may not have put anything smart on the USB side of things since it wasn't intended, so it might draw more then the 500ma of USB2 and damage a USB port. Probably could add a mosfet, schottky diode and pull down resistor to prevent back flowing the USB from wall charger if both were plugged. Likely also need a mosfet to switch the charger circuit from 500mA to 2A depending on which source plugged.
EDIT: Found a decent pic of what I was trying to say. Page 17 of the document
http://cds.linear.com/docs/en/datasheet/4081f.pdf