You can find answers in older studies, when DNP was first introduced as a diet drug, for example:
Cutting WC, Tainter ML. Metabolic actions of dinitrophenol with the use of balanced and unbalanced diets. J Am Med Assoc 1933; 101: 2099–2102.
This paper compared diets (high fat, high carb, high protein, and isocaloric) while looking at metabolism, nitrogen loss, acidity, and change in body weight, while on DNP.
"Dinitrophenol, used in doses of therapeutic range, caused increases in metabolism of the usual magnitude irrespective of the type of diet."
I know a few folks who have done CKD style while using lower dose ranges of DNP with good effect. So I don't think running a particular type of diet means fuck-all in the presence of such a powerful drug.
Hypothetically, if I was going to use DNP, I'd use it at 200mg/day for a longer period (2-3 wks), and run my normal CKD...I'd basically look to get the results I wanted by caloric restriction without elevating the dose.