Cannot run BLDC motor at peak current and low RPM
I cannot supply more than 1A and can only supply above 1A if I
  increase the speed.
  
  Why is that?


Motor speed is proportional to voltage. To run at lower speed you have to lower the voltage. But you only have a fixed supply voltage of 24V, so you are reducing rpm by lowering the PWM ratio in the controller, which lowers the average (effective) voltage at the motor. Why does this matter?

During the 'on' part of the PWM cycle the controller feeds supply current to the motor, but during the 'off' part it disconnects the supply and recirculates current through the motor. As PWM ratio (throttle level) is lowered the ratio of motor current to supply current increases in inverse proportion. At 50% PWM ratio the motor draws double the (average) supply current, at 25% PWM it draws 4 times as much. Thus at low throttle the current you measure at the supply is not the motor current! 

If your controller limits motor current then at low throttle the supply current will be less than the current limit. The peak supply current is also equal to the motor current, so your supply might shut down even though the average current (measured with a meter) is below the limit. 

Understanding why the motor phase current is different to the power supply currents