In addition to keeping you adequately hydrated -- which can also help alleviate constipation -- drinking lots of water can also help offset still another low-carb diet problem: bad breath. The ketones produced during the diet can lead to what is sometimes described as a fruity odor although it is often described as having an almost "chemical" odor similar to acetone or nail polish remover.
I know you posted a few months ago, but I thought I would reply just incase its still relevant. After having a daughter who LOVES veges, I then had my son who at a year old suddenly refused veges. It was getting so bad he would make himself throw it up if we forced him. But we kept on going trying everything we could while getting extremely frustrated (While also being told by everyone he was too small and sickly (Which he always was but got worse when he stopped eating the veges)
Some people on a keto or low carb diet choose to count total carbs instead of net carbs. This makes it more difficult to fit in more leafy greens and low carb vegetables (which are filled with fiber), so you should only try that if you don’t get results with a net carb method. And, start with reducing sugar alcohols and low carb treats before deciding to do a “total carbs” method.
Use fat as a lever. We’ve been taught to fear fat, but don’t! Both keto and low carb are high fat diets. Fat is our source of energy as well as satiety. The key to understand, though, is that fat is a lever on a low carb or keto diet. Carbs and protein stay constant, and fat is the one you increase or decrease (push the lever up or down) to gain or lose weight, respectively. So if your goal is weight loss, eat enough fat to be satisfied, but there’s no need to “get your fats in” once you’re satisfied.