Is there a rank requirement to be the approval authority for a DA5500/5501?

Misplaced comment...sorry.

"Is there a rank requirement to be the approval authority for a DA5500/5501?"
NO, there is not

It looks like there is no rank requirement. I was first "taught" at BNCOC but that doesn't mean it cannot be taught before. Whoever has been trained per the regulation would "prepared by" and that person's supervisor would sign as approver. In a theoretical universe an E1 could be trained to do it and an E1 with more DOR could approve it. In a real universe the 1SG or PSG does it. In a barracks lawyer universe, if you get taped by a trained SPC and a SGT approves it, it is legit.

AR 600-9
The individual taking the measurements must have a thorough understanding
of the appropriate body landmarks and measurement techniques. Unit commanders will require that designated personnel have read the instructions regarding technique and location and obtained adequate practice before official body fat determinations are made. Individuals taking the measurements will be designated unit fitness trainers, certified master fitness trainers, and/or trained in body circumference methodology, as specified in para 2–18c.

2–18. Designated unit fitness training noncommissioned officer or master fitness trainer:
A designated unit fitness training NCO or master fitness trainer will—c. Train other command designated NCOs in proper height, weight, and body circumference methodology to assess
body fat composition.