Seems like I need to know the impedance of the scanner REC output and the dock's aux input. Bearing in mind I have no idea what I'm doing, I turned to Alan Wolke (W2AEW) for help. He's got a load of great videos, one of which explains how to measure -- or at least approximate -- input impedances of unknown circuits with a scope, a function generator, and a variable resistor.

1. Dock Input Impedance

Test setup is the same as before: function generator making a 1-kHz sine wave at 60-mVpp into the mic jack of the Baofeng. Transmit into 50-ohm dummy load on 146.420 MHz, scanner receives and pipes it into the 3.5-mm extension cable and into the aux jack on the dock. Scope probe on TP5. I already know that unloaded, the scanner puts out 680 mVpp, so if I adjust the pot on the sidekick board so the signal is half that, that means the resistance from TP5 to ground should equal the input impedance. I got 18k ohm measuring it this way -- seems high given how much the dock seemed to load down the scanner, but whatever.