No doubt it varies by state, and likely even by which clerk you get at the DMV. But I've been through similar issues several times with the CA DMV and it's never been a problem. From their reactions, it appears to be a common SNAFU. The TR3A I bought here back in 84 or so was later found to have a totally different number than the one on the title & registration. No one noticed, until the state started matching insurance against VIN numbers. (My insurance of course had the number off the car, so it never matched.) Took the car to the DMV and told them the story; they issued a new title and registration.
With Stag #2, the VIN plate had obviously been tampered with, the plate on the car was the "European" size which is smaller than the original US plate and you could see the yellow paint that had been under the original plate. After a little hemming and hawing over what year to call it, they assigned a new VIN and created all new paperwork.
With the current TR3, I never got the original title, just a bill of sale. They never even looked at the bill of sale, just asked how long I had had the car in my possession (several years), had me fill out and sign a "statement of facts" outlining how I got the car and wrote up new paperwork. There was some extra hassle over keeping the old "black" plate (they wouldn't let me keep the plate on the car unless I could prove it had once been assigned to that car), but that didn't make any difference to the title process. If the previous owner's aunt hadn't found an old registration, the DMV would have just assigned new plates to the car (and let me sell the old ones on eBay).
Anyway, my suggestion is to go ahead and get the numbers sorted out now, rather than waiting until you want to sell the car. I doubt your DMV will be overly impressed with a certified Heritage Certficate, since it says nothing about who owns the car on the certificate nor does it prove the car hasn't been tampered. And in all 3 of the cases above, it took more than a month for the new paperwork to arrive, which might make your buyer nervous. I know I was unhappy about the tampered VIN plate (and felt a bit stupid for not noticing it earlier).