I doubt you can make things worse with dry ice, I'd say it is worth trying before you spring for a new door.
I suspect you could make it worse. It depends on how the door is insulated. If it was manufactured with polyurethane foam injected into the door ( used to be the standard method for maximum insulation ), then the extra cold ( -80C ) could shrink the foam away from the door outer and leave a void that could show as condensation on the door outer after a few weeks of operation.
My suggestion would be to approach the agents for a new door cost, and work from there. Perhaps start suggesting to your better half that the dent adds character, and the cost/risk of repair is disproportionate to visual issue. Other options are to suggest they stick something patterned on both doors that will minimise the visual effect and hide the dent.