What is the reg governing appointment of an NCO to conduct height/weight officially?

AR 600-9:

2–18. Designated unit fitness training noncommissioned officer or master fitness trainer
A designated unit fitness training NCO or master fitness trainer will—
a. Prescribe proper exercise and fitness techniques, according to Field Manual (FM) 7–22, to assist Soldiers in meeting and maintaining body fat standards.
b. Assist commanders in developing programs that establish a physical fitness program in accordance with FM
7–22.
c. Train other command designated NCOs in proper height, weight, and body circumference methodology to assess body fat composition.

AR 600-9 would be the source, but I believe it is just a designated fitness training NCO (usually the "training room") or a MFT (master fitness certified) NCO. There ha to be at a minimum of 2 people, 1 to place to the tape and 1 to ensure the measurements are taken properly.

AR 600-9 The Army Body Composition Program, App B para 1b.