Those cables should be rated around 13A @ 220V (unless the extension lead states a lower figure) but could struggle to deliver 20+Amps @ 48V without having a significant voltage drop across the length of the two wires.
I have found that lead acid batteries often suffer from a noticeable voltage drop under load, even when connected with good heavy duty wires. However, if they are connected with thinner wires that cannot handle the current flow, the overall voltage drop will be even greater.
Under heavy load, the combined voltage drop across the batteries and wires can sometimes be large enough to trigger the Low Voltage Cutoff in the controller, which forces the motor to cut out, and I suspect this may be the reason why your motor is cutting out under heavy load.
Weak battery cells in older batteries can also suffer from excessive voltage drop under load, but simply measuring the individual battery voltages before and after the event is unlikely to highlight the problem, as the battery voltage will usually rise again as soon as the heavy load is removed.
The best way to confirm whether or not the cutting out is being caused by a significant voltage drop at the controller is to measure the actual supply voltage directly across the controller's battery connections while the motor is under full load.
If the voltage drops below ~42V (for a typical 48V controller) the Low Voltage Cutoff will usually switch off the motor power to prevent the batteries from being overdischarged.
Using thicker battery wires (combined with good battery terminal connections) will greatly reduce voltage drop between the batteries and the controller, which should also prevent the Low Voltage Cutoff from being activated prematurely.
Alan