Following the links in this comment by @Uwe I've found some interesting sources of information on Ham (amateur) radio operators trying to listen in on Apollo transmissions. One of them is Sven Grahn's Tracking Apollo-17 from Florida.
Ham's setting up a 9 meter dish to receive signals from the Moon, and a doppler shift measurement (offset) of the received signal at around 2287.5 MHz as the spacecraft orbited the near side of the Moon. From Tracking Apollo-17 from Florida.
It looks like there will be at least 50 kHz of doppler shift for one orbit of the near hemisphere of the moon, eyeballing the graph.
50 kHz divided by 2287.5 MHz is about 22.9 ppm. Multiply that by the speed of light and I get a velocity change of 6557 m/s. Using a GM_moon of 4.905E+12 m^3/s^2 and an altitude of 60km, I esitmate the orbital speed of only 1650 m/s.
That makes the estimate of the change in line-of-sight velocity four times larger than the orbital velocity. I can understand twice, since it changes from coming towards to going away. But not four times.
Question: What is the explanation for such a large doppler shift?
Images from Sven Grahn's Tracking Apollo-17 from Florida