There will be common mode current flowing on the coax, through your un-grounded radio and its power cable to wherever you are getting negative battery terminal. Some of this current also flows on the positive lead and microphone lead. It is a far from ideal situation. I would run a short jumper from the radio chassis to the vehicle sheet-metal before doing any antenna adjustments.
Grounding the radio chassis won't do anything to solve common-mode RF in the coax. That has to be fixed at the antenna end.