does a 32 dord sedan have a model name??

For the insurance, I'd go with "Tudor" if it's a two door sedan or "Fordor" if it's a four door. Those were the body style names Ford used for its different models (Standard Coupe, Deluxe Coupe, Sport Coupe, Roadster, Phaeton, Tudor, Fordor, Victoria, & Cabriolet). These would be more or less the equivalent to today's model names. These names would satisfy the insurance people.

Edit:Dad's 1932's solution would be a good way to go, too.

 

I'd take them the title and reg. and get the HMFIC to do the paper , if you just call it by what ever "brand x" name that isn't correct , they will just tell you to pack sand if you ever have a claim , then who pays for the damage and or injury's associated with the mishap, promise ya this , it wont be the insurance co.

 

You need an insurance that specializes in classic cars, other wise if it ever comes time to make a claim you'll have the same problems you're having now.
If it's a hotrod, make sure they know, if it's a replica make sure they know! Most '32 Fords aren't from 1932 and aren't built by Ford. (Forget that little one, and you could end up doing jail time. Someone here did)
Insuring a hot rod is way more than clicking on drop downs. You'll need to submit an engineers report and a recognised valuation, if you ever want to see any payout after a fire of theft.
That sweet girl on the phone isn't the one who'll be paying out the $50.000 if someone nicks your car. I'll be some bastard in a suit who'll visit you and his job is to get his employer off the hook.
Ed