gauge is likely easy to change. Not cheap, but Amtrak demands expensive inspections and refurbishment to run, so the cost of changing the gauge is likely fairly small compared to the other costs.
This is actually quite a significant technical achievement - for example, a similar project in Japan failed.
Japanese Railways wanted to build a train that can run at full speed (~300 km/h) on the standard gauge (1435 mm) regular Shinkansen lines but also use the narrow gauge (1067 mm) existing lines at slower speed. Those older lines would not have to be rebuilt for the Shinkansen standard & there would still be significant time savings:
This failed to produce a viable train, resulting in falling back to track rebuilds or using relay trains that connect directly from Shinkansen to the local rail line on the same platform.
I wonder if the slower-speed lines have looser tolerances.
Then maybe running the gauge-change train on the slower lines might kill the train's tolerances before it moves back to the super smooth high-speed lines.
There's a station on the main line that loads full sized cars with tanks on them onto little bougies that take them up into the mountains for training.
No, it isn't easy to change at all. Not unless the car was specifically designed for it, and not nearly that much of a jump. The ones that exist in real life are almost all for switching between Russian (5ft) and Standard (4ft 8.5in) gauges.
To be eligible to run as part of an Amtrak train any car must past all FRA rules/guidelines, which a Euro-spec car absolutely will not without hundreds of thousands of dollars of work.
It would be MUCH cheaper to start with a car already in the US and meeting those standards. Much, much cheaper. Still not cheap, but in the realm of the practical.