Hi and
to the forum.
It's difficult to say whether the problem is the motor controller, battery,
BMS, key switch or wiring.
My suggestion would be to test the battery under a small load using a high wattage mains voltage light bulb to see if the battery voltage drops like it does with the controller connected.
A 240V 100W bulb should draw about 0.2 Amp (~10Watts) from a 48V battery, which is not very much, but it's still a lot more than a good controller should draw under a no load condition.
If the 100W bulb glows dimly and the battery voltage does not noticeably drop, then I would say the problem is either the controller or the wiring between the controller and the battery connector which is causing the problem.
If the battery voltage drops below 52V when just the bulb is connected, I would say that the battery itself is faulty.
Hopefully this simple test will help you to identify the source of the problem.
Alan