Yes, if the base is above ground, it will affect the impedance.
Consider that the feedpoint impedance of any vertical (any length, any
diameter) fed against a perfect ground plane in freespace will be
exactly half the impedance of the same vertical fed against a mirror
image of itself in freespace (in other words, a symmetrical dipole).
The closer the base is to being a ground plane, the less effect there
will be because of its height. How big is that car roof compared to a
wavelength of the frequency you're interested in? Since the impedance
of a halfwave fed against whatever counterpoise you have will depend a
lot on the diameter of the halfwave element, you'd probably better just
tune your matching network to get acceptable match to whatever you're
using to feed the antenna.

If there's no matching network between your "return loss measurer" and
the antenna feedpoint, I'd expect a better match for the quarter wave
than for the half wave, assuming an instrument calibrated for 50 ohms
(or even 75 ohms). A better match would be a higher return loss. Are
you interpreting the numbers correctly? What do you get when you put a
50 ohm load on your instrument? What do you get when you put a 1000 or
2000 ohm load on it?

Cheers,
Tom