Although I can’t provide patient specific advice as each patient is different, in general if the physician and family have selected Gelmix as the thickening agent of choice, then they would use it for all liquids including water, not just the bottle. The consistency is helpful especially if a problem arises and the physician would then know the product in question.
If the child just started pureed food, then gradual introduction of one food at a time is recommended for allergy concerns. Usually the doctors start with infant cereal. Once he/she eats several pureed foods, some are naturally hydrating like applesauce, mango puree, watermelon puree, depending on oral phase of swallow function. If not tolerating thin puree, can thicken a little with mashed potato flakes or infant cereal.
You should also seek further support and questions directly to your physician or healthcare team.