The spies from CIA had this problem solved long ago.
You need to have the relative dielectric constant between the coax shield and the sleeve equal to the dielectric of the cable. By this you maintain same velocity. Also you want to keep the impedance same, which requires quite a thick layer ...
It is more than 15 years ago, that I've done it.
But if I remember correctly, the 3D screenshot you're showing is quite misleading. The pattern is drawn around the 1st antenna / center of coordinate system.
Delete the 2nd antenna and compare the gains to make sure nothing changes at all.
Some (in fact quite a few) mobile antennas are effectively an "end-fed" design rather than a "ground-plane" design. One way to identify them is that they will generally have a noticeable transformer coil at the bottom (usually enclosed) to transform the impedance. If an antenna's description includes the words "ground-independent&...