It occurs to me that if you mounted the car on a pivot, the whole assembly wouldn't be able to stand up on its own. Of course, you possibly could still use the bike's side stand. But it would be sort of like the suspension in a car (or a springer fork) - without the spring and damper, the whole thing collapses. If you wanted it to be self supporting, you'd need a spring system to allow the movement while supporting the assembly. Maybe torsion springs on the pivot. You'd probably have to have two opposing springs so that it returned it to a vertical position normally. A damper might be necessary if the system is too bouncy, but maybe not.
Also, the side car wouldn't remain truly level; the geometry might be kind of odd. Assuming it was mounted on the right side, if you lean left, the pivot point on the frame will raise up to an extent, and if you lean right it will drop. Probably not a lot, but some. How much would be based on how wide the pivot extends from the bike's centerline, and far above the bike's roll center the attachment point is. Since the roll center is at the tire contact with the ground, even mounted to the chain stay it would be a foot over the roll center.
But aside from all that, there's no reason it wouldn't work. Just make sure that you've got enough clearance for the pedal swing when you're leaned into the car.