The time dilation experiment involves measuring the mean lifetime of the muons ... the LED is most likely for calibration.
Can light from the LED get to the scintilator and can it get to the photomultiplier?
Note:
The pulsing light was accidentally turned on by someone and caused mean lifetime of about 1/4 of the expected value.
... this statement is nonsense: you don't know that the LED has any effect on the muon's mean life and you do not know of any mechanism for this to happen either so don't say this. What you have written here is a conclusion when you should be writing observations.
Say only what you know first - then speculate.
What you know is that you got a bunch of signals from the equipment.
If you assume that all the pulses come from muons, then use the data to work out the muon mean-life, you get a low number.
However, some pulses come from the LED by some mechanism that depends on stuff you have not told me about how the equipment was set up.
You could maybe rescue the experiment by using what you know about how the LED and equipment interact (which you have also not told me) to take account of this as a kind of background pattern. ie if you know the LED pulses at 1Hz you should be able to remove the resulting artifacts with reasonable confidence.
But if you don't know how the LED interacts with the rest of the equipment, then the experiment is a wash. You have no useful data.