Never tried to quantify it, but I don’t doubt for a second that any significant change in the headstock could have some impact on some aspect of the sound of your bass.
Little bit of background - a few weeks ago I was playing my second best sounding bass (Capped Goth) and got frustrated with the crap stock tuners, so I pulled out a new set of Gotoh Batwing tuners and installed them on the Goth. Put the strings back on, and Crap, a dead spot on the D string that was never there before. Took the G and D tuners off and swapped them, dead D still there in the exact same spot (so it’s not the specific tuner). Put the original crap stock D tuner back on and the dead spot is gone. Both the Gotoh and stock tuners fit the headstock holes equally well (so not a slop fit difference).
Apparently something as simple as a slightly lighter tuning machine can impact the resonance or vibration of at least the D string on the neck, directly effecting what the pickup sees magnetically and the resulting sound the bass puts out. Could a significantly different weight headstock have an impact? Quite possibly. Would it be noticeable, who knows. Can it be quantified, wood has too many variables to make it worth quantifying, just mark it down as a known attribute of varying consequence and move on.