I have the same car and the Panasonic ZS7, which I will assume is similar.
First, set the camera on "P" for Program. Go into the menu and under the regular photograph menu (not the video menu), scroll down to "Metering Mode." Select the spot metering option, which is an icon of a single dot in the middle of the screen. Exit the menu and hit the record button when you're ready. Make sure you have the camera in P when you record. If you put it into Automatic, it will use automatic metering and look bad again.
Set up like this, I get a clear view of the road. I can see myself shifting, but the gauges can't be seen because they will be too dark.