If you increase the battery Ah then it will be more happy at a higher discharge rate, IE: 10Ah rated 1C is OK for max 10amp discharge, change the battery to a 20Ah 1C then its OK for 20amp discharge....
The battery is 1C in both cases but the larger one will deliver more amps... The control is the gate keeper as to how much current is drawn from the battery so you choose the battery with an eye on what the controller is rated for... A 15amp rated controller (quite typical for 250W motors) will pull more or less 15amps from the battery therefore it makes sense, so as not to stress your battery too much to use a 20Ah battery if the one you are thinking of is rated 1C...if the battery is rated 2C then guess what? you could use a 10Ah one.
Pushing more current through the motor will increase torque up to a point but put more strain on the motor and if fitted with internal gearing put the gears under more stress. More heat will be produced in the motor and the controller will also be under more strain....there are limits and 15amp is a safe reliable one for Typical Bafang motors. Also current limiting in controllers can be imprecise, I've measured two different 15amp rated controllers and both pull 20~21amps from the battery, so keep this in mind when looking at batteries.