Hello. Here are the parts of your story, early on, that I think there's a problem with:

Now it just so happened that a man named Ben lived on the river halfway between the two cities. In fact, the river ran right through the center of his property.

One day, he decided he'd leverage his ownership of the river to make a little cash

So he built a wall across the river, and put one little gate in the middle.

The implication is that the protagonist would face no legal repercussions for building the gate. That assumption is not warranted.

This article from FEE is instructive. Here's one of the relevant paragraphs:

The other doctrine, which prevails in most western states, is called the appropriation doctrine. According to it, water rights may be acquired by either riparian or nonriparian landowners on a “first-come, first-served” basis. The first to make “beneficial” use of water acquires the right thereto, and in case of water shortage the rights acquired later in time have to give way first. This is not a system of pro rata distribution but a system of priorities in which prior (in time) water rights are entitled to be fully satisfied before subordinate rights can claim any water at all. These appropriation rights are rights to use certain amounts of water during specific periods of time for certain purposes. The holder of the right cannot ordinarily change the use without losing the right and cannot sell the water to someone else to use for any other purpose at any other place. The continuance of the right is dependent on exercise of it. Failure to use the water for a period of time results in forfeiture of the right.

Here's Murray Rothbard talking on the subject.

It is immediately clear that the route to justice lies along the appropriation rather than the riparian path.

Rothbard was an advocate of the appropriation doctrine. So at least to the extent that the hypothetical ancaps are Rothbardians in this respect, it seems likely that this approach to water ownership and use would be the one used on the island.

Under this doctrine, the barge operators have already homesteaded the right to use the river for their operations

Ben, by setting up his gate, is interfering with the exercise of that right. He should expect a legal challenge to his actions.

Perhaps the real moral of the story is that if you think of a scenario that seems like a difficulty for a voluntaryist order, it's a good idea spend a little extra effort to search for solutions that thinkers in this tradition have already proposed.