A 16 wk old pup, that drinks a 1/4 cup of water, and a few ice cubes between meals, IS NOT OVER DRINKING. It is dangerously under drinking!
Diarrhea is not caused by drinking too much water, it is caused by immune response to a trigger, either a pathogen or a allergy or stressor perhaps. I"ll come back to this point.
Never restrict a dogs water, unless it is in kidney failure, and has low potassium levels, and then and only then, restrict water. Or if its about to undergo surgery. See, extreme scenario's only. HOpe you get my drift here.
Here is my opinion as a medical professional of your scenario, and the link between water, and runnier shit.
Your dog has diarrhea. You do not know cause. IF you reduce fluid intake into the diet, the waste continues to have moisture taken from it, from tummy to ass hole. The less water, the less runny.
So it seems to make sense, give less water perhaps.
But all you are doing is masking diarrhea with dehydration effects. and it is dangerous what you are doing.
Please consider adding free access to fresh water 24/7, 365 days a year. If the result is emergence of diarrhea symptoms, address the cause, not add a risk factor to diarrhea in a pup by removing life saving water.
Same food, same result for 16 Wks. And you have done the right thing in changing diet.
Dont forget free access to water regardless of diet, ever again please.
Raw is great. Expensive, but great. So, likely food stuffs to upset a dogs tummy allergy wise are:
in order of trigger likeliness:
Grains
Beef
Chicken
What does that leave to feed a dog? Remember, you are after protein. So think: pork, lamb, offal, goat, fish, camel, kangaroo.
An allergy becomes an allergy, when either 2 things have occurred:
1. a allergic reaction test has been completed, and confirmed via a vet.
2. a elimination diet has proven what the trigger is.
Outside of these 2 scenarios, you have a guess. No more, no less.
Here are some other causes of extreme thirst for instance, diabetes, toxin ingested, heat. But like i say, a 1/4 cup of water daily, is dehydration, not polydypsia < over drinking.
And, do your pup a favour, go find a vet that knows what they are doing, your's clearly is rather suspect. And had not even suggested you should NOT restrict water in a pup with runs, is negligence in extreme!