I'd say forget trying to tamper with it. Your best bet is to try
reprogramming
the reciever in the overhead motor controller. If it's of recent vintage,
that would make more sense and easier.
The slug is probably part of the RF tank circuit, so detuning it could
cause freq. drift and gets you away from a fix.
A passive transmitter sounds impossible, as an LED indicates curent
consumption......you have a battery !
I would venture that the LED lights, but the transmitter is too weak to
effectively radiate the signal. It happens. Replace the battery before you
twiddle anything else. If you adjusted the RF coil, you may have doomed the
frequency setting. You may have also somehow altered the DIP settings
if the unit uses digital encryption. (ie dropped it and reset a DIP)
Do the following:
-replace the battery, clean contacts and try again.
-check the motor housing for the learning mode. "Teach" the receiver the
code for your transmitter by following instructions. Usually text on the
box tells you how to teach it.

If in fact you transmitter went bad, you need to buy a replacement, and
teach the motor unit the code anyway. Do a search for your brand online.

Good luck..... Les KA9GLW