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 half-width 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