It would have been exceptionally difficult to fool the ground station operators.
The PM transponders were 'phase coherent'. In "1-way" mode without an uplink, a local oscillator generated the nominal downlink frequency so the ground received frequency would depend on both Doppler and oscillator drift. When an uplink was received, the transponder automatically locked on and transmitted at exactly 240/221 times the received uplink frequency. This changeover was very obvious to the station operators, who aways observed it during station handover. Any difference between the expected and actual frequency was entirely due to Doppler, which directly indicated range-rate. This Doppler was integrated and used to update the range measurement.
To initially calibrate this range at the beginning of a pass, the ground turned on a PN (pseudo noise) ranging signal that was turned around by the transponder. The PN signal degraded the overall signal-to-noise ratio so it was turned off as soon as the range was acquired. It didn't have to be used again until the next handover unless lock was momentarily broken.
Only one ground station could transmit at a time, but any number could listen. Being in different locations they'd hear different Doppler shifts.
To even have a chance of fooling the ground operators you'd need real spacecraft with real transponders that worked like this. But several of their functions were manually controlled by the astronauts on request from the ground, e.g., antenna switching, selecting wide/medium/narrow beamwidth for the CSM high gain antenna, manually or automatically pointing the CSM and LM high gain antennas, enabling or blocking the path from the uplink command receiver to the computer, and enabling PN ranging turnaround only when it was actually needed to prevent noise from being retransmitted. On occasion things wouldn't work right (the CSM high gain antennas frequently misbehaved) and troubleshooting required voice conversations with the crew.
Usually the Capcom relayed the ground's requests, but on a few occasions the ground station operators spoke directly with the astronauts, e.g., when the link between the station and Houston was broken. For obvious reasons this wasn't carried by the TV networks, but the ground stations made their own recordings.