Does the AMA have anything equivalent to SCCA Autocross Solo II?

Subject says it all. SCCA Solo II racing is fun and safe. You can do
it in the family sedan. Only one car is on the course at a time.
The course is typically in a parking lot and marked out with cones.
It tends to be quite tight and almost everyone runs in second gear.
You get several runs and your fastest run becomes your final time.
Fastest time wins. Simple.

Autocross has some speed, but not a lot. I would guess 30mph is about
the limit. The link above and the Japanese course are definitely low
speed exercises. As the owner of the Parking Lot Moves site points
out, crash bars and mirrors are the usual victims of such a course.
I'd just take the mirrors off in advance, and use a naked bike with
custom crash bars - maybe a Honda Hawk or a dual purpose bike.
Oh, and good hip & knee armor in the trousers.

The brilliant part of the sport is that it is cheap. You also do not
need to be a member of the SCCA or have an SCCA license to participate
in Autocross.