Max, without knowing all the current needs (pun intended) and the future electronic upgrades I would suggest using the small emergency light SLA batteries. They come in both 6V and 12V flavors, are reasonbly priced and availible at most hardware stores and battery specialty stores. The advantage is if you need more power in the future increase the AmpHour rating on the batteries or add more batteries (in series). I would power the lower than 12V electronics off the simple regulators: 5V or 3.3V with some filter caps. This simplifies the electrical supply side and makes it easier to avoid issues with low electronics batterys and fully charged motor batteries aka: "Why won't my robot move??" I would also suggest adapting a battery charger with quick connect/disconnects and wiring these up in parrallel with the batteries to allow for simple in robot charging. If you bury the batteries in the base then either in place charging is required or ease of access.
One thing I know we talked about and I will mention again: main line power switch with a fuse or circuit breaker that is easy to "get at" from outside the robot. This is both a safety and a convenience thing. Safety in that when you hit the power switch the robot is OFF and also to prevent shorts from burning up motors/batteries etc.. Having a breaker panel to distribute the power to the motor controllers is also a good idea.
I'd say screw it and just wire up 3 seperate chargers to save time. If my target was building a robot to play with programming and robot behavior, I wouldn't want to spend my time designing optimal charging circuitry.
Then again, maybe the optimal design of the circuit is part of the fun for max.
Commercial LiPo multi cell battery packs have balancing connectors incorporated in them. These are monitored by the charger for each cell and the amount of current charging each cell is varied according to the state of charge of each cell. In this way each cell is optimally charged. The better cell packs are built from matched cells hence optimising the charging / discharging system.
Nickel Metal Hydride or Nickel Cadmium cells as used for model car racing are another option and again are available ready assembled. Automatic chargers are also cheap and both readily available from your local model shop.
I see that the free run current is about 300mA but the stall current is 5000mA. The LiPo battery being considered is rated at a maximum discharge of 2000mA so one could anticipate problems in the system voltage collapsing under stall or high acceleration conditions. This is not a high discharge capable battery, it has built in self protection unlike the aircraft types which are still stable at 20C.
As an aside at 200rpm and 4" diameter wheels the speed would be (200 * Pi * 4) / 60 inches per sec = 42 ins/sec = 3.5 ft/sec = 2.4 mph which seems reasonable.
Actually, the Adafruit page says "This battery has a capacity of 2500mAh for a total of about 10 Wh." Now Power (in Watts) = Current x Voltage, so what does "2500mAh for a total of about 10 Wh" actually mean
LiPo batteries are built in increments of 3.7v per cell. So 3.7V * 2.5A = 9.25 Wh
Electric model aircraft commonly use three cells in series giving 11.1V and in capacities of 500mA / 1000mA and 2200mA. These cells have common chargers which cost about £20 whilst the cells cost about £10 to £20 each set. The LiPo cells can be loaded up to 20 times thier rated capacity without damege. I have used them for years without any problems : just don't damage them! Your local model shop will show you what is available.