Alright, I thought I was finished rigging the car, but thanks to Chirone's help, I spent some more time on it and got it to a much better point.
I don't know where the error came from before, because I was quite easily able to add two parent constraints to the same geometry. Then I used driven keys to switch the weights from one point of rotation to the other based on a custom attribute for my main control. Very nice
I also fixed up my idling control so it now idles away when I turn on my idling boolean attribute. (I wanted the car to move up and down very slightly) No manual animation for that, just added an expression based on the sine of the time. The extent of the motion needs tweaking, (it's going like a jackhammer) but it works.
Thanks again, Chirone!