6. Battery:36V10.4Ah lithium battery, Removable battery
there is no way you can get the best out of the 36V 250W G06 with this battery leave alone using it with the 500W G06. You can also damage the battery easily when using a low capacity battery.
The Chinese grade their motors on the basis of continuous power. If it says 250W on the label, you can pump 250W into it all day, the mass of the motor will dissipate the heat without burning the coils or the sensors. 350W motors have thicker wires, lower stalling resistance. 48V 500W motors have thicker wires and usually have more turns in each coil so gives higher torque at low speed and better acceleration.
Typically, when the motor is labelled as 250W, the maximum peak power is double that. Your controller and battery need to be paired to the motor so that they work well together. For example, 250W motor can peak at 500W, the controller should be 14A-15A 36V. 350W motor peaks at 700W, controller should be 18A 36V. This is a popular choice for manufacturers, they simply ask for 350W grade motors for use as 250W.
500W motors are suitable for offroading where you would expect high, short gradient, high acceleration. The controller would be 48V 20A.
If you have a 36V 15A controller, then choose a battery which does not get damaged at that current continuously. That's belt and braces. As a rule of thumb, 1.5C. If C is 10AH for the battery, then 1.5C is 15A for the controller. If your motor is 250W then 36V 10AH is the minimum required.
If you want 500W motor then you'd need 48V 14AH.