Sounds like alternator whine to me too. Or perhaps one other problem.
It might be the car's system though, generally the radio pulls more current when transmitting and this will increase the alternator whine.
Now, from the sounds of it the metal plate on the roof is only "Grtounded" via the antenna, This is not a problem. You must have a ground connection on the radio, (DC at least) or it would not work.. YOU could try a ground strap. not to the engine but to a handy point on the frame under the dash. But I do not think it will help. Use fairly heavy wire. The problem is, if you have more than one ground, you can set u what is called a ground loop. and that, too, can cause issues.]
The other thing is the mic gain if the radio is adjhustable.. If it's too high bad things happen, but not, normally, a pig squeal.
There are other possible issues too, Mag mounts are not the greatest at HF frequencies, they do not really shine till you get to VHF or higher.
Home is where I park it!