A digital multimeter doesn't offer much of a load. Testing the gray wire with an incandescent 12 v test light would probably drop that 3 volts to 0 volts.
Since the car originally didn't come with Homelink, the BCM probably isn't putting power on that wire. It may be possible to enter the Homelink sales code (CHP?) into the BCM with the scan tool to tell it that it that it now has Homelink.
Or simply cut & splice an available 12 volt source into the gray wire.
When I install a Mopar Remote Start system in a vehicle that didn't come with Remote Start, I have to enter the Remote Start sales code into the module to tell it that the vehicle now has that option. After that, everything works.
This is the new way to add different options to vehicles if the wiring & other required foundation is there to support it.