Even though a bowling alley may be required to provide drinking fountains, if there is beverage service as part of the business, then a drinking fountain is not required by Texas Plumbing Code 410.4. But in that instance, the restaurant is required to provide a container free of charge. (See code excerpt below.)
In this case, the bowling alley has provided the fountains, so they are under no obligation to provide a free container. They have also provided a dispenser, which is not required by code. The Owner has the right to limit use of the water bottle filler or to charge for its use, but it sure seems silly to not bottle filler use when there is a fountain bubbler 12 inches away.
I don't know you at all, so don't take this personally, but I'm curious: if I were to ask the Owner, would they say there is some other reason why they want to discourage you from using bowling alley facilities?
Are you a paying customer for bowling, video games, food or snacks, etc.? Do you do any kind of business transaction with them?
Are you a vendor/delivery person to the bowling alley, refilling you water container before heading out to your next delivery?
Or are you a person from the outside who only goes into the alley to make use of their code-required plumbing facilities, shelter, air conditioning, etc. but you don't do a business transaction?
Where restaurants provide drinking
water in a container free of charge,
drinking fountains shall not be required in those restaurants. In other
occupancies where
drinking fountains are required,
water dispensers shall be permitted to be substituted for not more than 50 percent of the required number of
drinking fountains.