So you your remedial request is for low BMI and they have asked for a note from your doctor that says "Provide note from private physician documenting no history of eating, endocrine, or genetic disorder." If that is what they have asked for that is exactly what I would do. You may still get DQ for being below the minimum weight and the note is to add documentation to your file to make a waiver easier. From USNA "Obesity will not be waived. If you exceed the body fat percentage standard on Induction Day, you may be denied admission to enter the Academy" So they treat obesity seriously, but, more lenient for being underweight.
Any way you can add an additional 5 pounds? Heavy combat boots lol. You could possibly get re-weighed and height measured to see if you could meet the standard. Long shot if that would work.