12%??
No. Or at least, not unless you're talking about canned or raw food (which then translates to about 40% by dry matter basis).
But for a dry dog food? I don't think there is any such food in existance to be so incredibly low in protein content. And if there were, it *could not* be labeled as adequate nutrition for any animal. The minimum (that is *minimum*, not ideal) for adult maintenance is 18%. Go below that, and the food cannot carry the claim that it is adequate and must be labelled supplemental only (which means you need to be feeding something else to make up for the inadequate diet).
If there is nothing wrong with your dog physically, you might instead try feeding a more nutritious food. It
isn't protein that causes excess weight, after all. Protein builds muscle. The major culprit for building FAT, in dog foods, is actually carbohydrates - for which dogs not only have no evolved need, but also little ability to deal with. Carbs thus get broken down to complex sugars and stored as fat (one of the major reasons many dogs on so-called diet or senior foods are overweight). What you're feeding currently is extremely high in carb content. What you're proposing is worse.
Incidentally, the minimum allowable carbohydrate content in dog foods is zero

(i.e. there is no minimum). Makes it even sadder that many of them are 60-95% carbs.
IMO, you'd probably have a great deal more success if you dump the low protein/high carb diets and swap to one that is high protein/moderate fat/
low carb - such as Innova EVO, NV Instinct or Orijen.