[...] use a 7.2 volt battery pack instead of a 6 volt one for the $50 robot using a resistor to lower the voltage to 6 volts? Right, I haven't missed something?
You want a steady voltage, not one controlled by the current draw.
If you use a voltage regulator in the circuit, you don't need to lower the voltage ahead of it, as the extra cell won't be a problem for the regulator, but unless you use a switching regulator, the extra cell won't give you any advantage over a 5 cell battery (but a bit more weight to drag around).
So I have three battery packs, one at 1400mAh one at 3000mAh and one at 1500mAh, which one would be best to use, and which resistor should I use?
The larger capacity, the longer the runtime, but the more weight as well.
If your motors/servos can drag it, go for the 3Ah battery.