Falcon Heavy is the cheapest kg to orbit rocket in the world by far right now and the largest one.

Super Heavy is only a first stage booster. Combined with the Starship upper stage this system is also called Starship. This system is still in development and has not yet made orbit.

NASA does not just pick some rocket for a huge expensive payload like that. A rocket needs to go threw a complex certification program, have a proven flight record and be well understood by NASA.

While SpaceX hopes to be flying with Starship by the point this would launch, NASA simply doesn't work in those time lines.

Falcon Heavy has flown once for a demo flight and was a decade late (iirc). Is that enough for it to be vetted?

It's flown a total of 3 times, including the demo flight. Besides the demo it's flown one commercial payload and one flight for the DoD.

Superheavy doesn't have an upper stage yet or even a complete first stage so if they're trying to start soon Falcon Heavy is the most economical option.

To nitpick Superheavy's upper stage is Starship, so it has an upper stage in testing, and the first stage doesn't exist yet.