Hi,
Here's what I know about importing into the US based on bringing a car in from Canada and when I researched about importing one from New Zealand back at the turn of the century:
The vehicle needs to meet Federal DOT and EPA standards (If you're in California, you may need something more). For the car I imported from Canada, I had to have proof of both of these to register it with my State's DMV. With the vehicle from NZ, the car would have had to go to a DOT-approved shop and have work done to meet the DOT and EPA standards. But, in both of these cases, this was for new cars….
From my readings, since your car predates the EPA, there are no requirements to meet. For DOT regs, I think the only one it needs to meet is that it has seat belts.
Last year, I was reading tons about exporting a Bambino from Europe and nowhere did I read where the motor had to be original. No one who imported into the US mentioned this. I wonder why this would even be a requirement? If it's needed, maybe there is a waiver for antique vehicles….
Also, maybe another member can answer this one: On the plate to next to the petrol tank in the boot, isn't the engine serial number stamped on it (Mine's too corroded to read it)? The number, if the motor is original, would be stamped on the engine block.
It may be worthwhile to Google: "importing a Bambino into the US." There are a bunch of people on this and other Fiat forums that have done this. Send them a private message and ask them about the process.
Also, I'd call/email another importer who does a lot between the UK and the US. They may be able to provide some more info.
Lastly, the Fed's DOT website may have helpful info on importing an antique vehicle into the country.
Good luck,