Suction Cups on Paint?

Suction cups on clean paint is fine.

On dirty paint it will work like a sanding block, giving you much greater leverage to grind the dirt in and scuff the paint.

 

Tie a short extension string to each end, then attach the suction cups to the door windows.

 

They won't wreck, but can mar painted surfaces.
One key is to make sure the cups and the surface attaching to are 100% clean & pristine.

For research on the marring, one could search Harley Davidson suction cup rider pillion seats on some Harley forums.

FWIW if silver or white paint, I would not be too concerned if you follow my advice about pristine surfaces & cups.
If black, well, you may see marring, yet a good machine polishing upon resale may work well to remove the marring.

Good luck in your decision.

 

If the length of the windshield cover is long enough, why not use the suction cups on the interior of the windshield?
If the tongue/suction cup hole area are not long enough to reach the inside of windshield, will the tongues wrap around the A-pillar? You just close the door and have the door seals hold the cover in place.