If the maximum lean angle of your couch doesn't surpass the tire's ability to flex, then there is no problem. In fact, you'll have way more contact patch with the car tire while its flexing.
A Harley that I got stuck behind at Deals Gap yesterday had its cornering clearance limited by its highway pegs hitting the ground long before any traction limit of even the crappiest tire could be reached. When that forced him to run wide in a corner and almost go off the road, I passed him (with my sport compound proper motorcycle tires and no cornering clearance issues) and left him far behind ...
I don't see any way around a car tire causing the bike to want to stand up in corners and require a fair bit of steering effort to get it to turn. I replace motorcycle tires that get flat-spotted long before they even remotely resemble the squareness of a car tire.