Can you still request an assignment on a TRADOC base (such as Fort Huachuca) if you were disqualified by TYPE II disqualification?

You can still be assigned to TRADOC bases in a slot that is not considered a "position of trust". However, depending on your MOS and rank, there may not be an open position at the Base you want, outside of the school.
Instructor is one of the positions of trust, so you would not be eligible for that.

The NCOA is very selective about instructors and won’t put a SGT on podium, as best as I recall. I believe the lowest they want are SSGs with a few years TIG. Calling branch will only get you to Huachuca, they can’t pin point you. You will pop up on the gains roster and the CSMs will divide up the inbounds. I’m a chief instructor (not at the NCOA); feel free to message me or look me up in global for more information.

There any plenty of Training Developer positions at Fort Huachuca that are outside instructor and drill positions.