I picked up a Velorex sidecar about 10 years ago for $600. Had all the parts including top etc. I had it mounted on a 71 Triumph but that struggled too much with it so I found a 78 750K for $450 that just needed carbs cleaned, tires, seat cover, and a repaint. I also put new chain and sprockets and geared it down a bit. Works great as a tug for the Velo. I've added 75lbs of balast to help keep the Velo on the ground during right hand turns. Setup and alignment is key to making it livable going down the road. I'd hate to have it as my only bike but it's a lot of fun as a hauler and round town errand runner.
I re-painted the whole works when I had it torn down so it matches up quite well. I have the car brake linked in to the rear of the bike via a cable. That really helps with stopping. Other than that it's been quite a reliable rig, and I can sure haul a lot of crap in it. Many trips to the Home Depot etc. or grocery shopping.
Picking a car is tough. too big and heavy and it's just cumbersome, too lite like the Velo and you can't keep it on the ground. So I opted for lite and balast that can be removed for passengers etc. The balast resides in the locking trunk behind the car seat. I keep the key for that in the tool tray under the bike seat. So the only key I need to carry is the bike key.
the 750 handles it quite well though, acceleration is good considering the extra 275lbs the rig is carrying. A dual disk front would be helpful but as much as I use it it's ok.
I think the perfect bike for a car would be the 750A. If I found one locally I'd pick it up in a heartbeat for this purpose. That would be just ideal IMO