Sorry for the confusion  I did mean Green beat frequency, and I had neglected the factor of 2 in my earlier calculations. However, the fit parameter "c" in my fit was actually the halfwidth at half maximum and not the full width at half maximum. After correcting for both these errors (new fit is Attachment #1, where I have now accounted for the factor of 2, and the X axis is the IR beat frequency), I don't think the numbers change too much. It could be that the frequency counter wasn't reading out the frequency correctly, but looking at a time series plot of the frequency counter readout (Attachment #2), and my earlier trials, I don't think this is the case (38 MHz is a frequency at which I don't expect much systematic error  also, the offset was stepped from 3 to 3 over 45 seconds).
The revised numbers:
Fitted linewidth = 2*c = 10.884 kHz +/ 2 Hz (95% C.I.)
FSR of Y arm (from elog 9804) = 3.9468 MHz +/ 1.1 kHz
=> Y arm Finesse = FSR/fitted linewidth = 362.6 +/ 0.5
Total round trip loss = 2*pi/Finesse = 0.0173
