No Stupid Questions

I'm going to try answering the titular question, despite the factual situation being wholly divorced from sanity, through no fault of the OP directly. Unless some new convention on the laws of the sea has been enacted vis-a-vis international waters -- which the Strait is -- a nation which provides some sort of service over that area is not entitled to remuneration. It would be gratis.

That said, in the very different context of aeronautical navigation, there is indeed precedence for providing a service to an international zone, but this is agreed to by multilateral treaty, not imposed. For this, we look to the massive area of the north Pacific Ocean, for which there is essentially no radar service available. And yet, to enable civil aircraft to fly without colliding with each other at speed, somebody needs to coordinate the flight routes and provide weather information. Through ICAO, the USA is the contracting state that provides such service over the Oakland Oceanic Flight Information Region. This means most of the Pacific Ocean is not USA airspace, but does follow the American procedures for organizing traffic (which follows from ICAO rules but possibly with small tweaks). This is the distinction between "USA airspace" and "USA controlled airspace". ICAO chose the USA because no other country can realistically perform this service, nor has anyone else put in an offer to ICAO.

To pay for this service, the USA FAA charges overflight fees. Since the FAA is a civil agency, it does not have authority to order a shoot-down of an airline's jets due to non-payment. Rather, the FAA can collect the owed fees through the American courts, just like anyone else would through a lawsuit for money damages. The FAA could also take adverse regulatory action, such as providing less or no service, subject to minimum obligations required by ICAO. The agency can also choose to cancel or disapprove of flights headed to/from USA airspace. If an airliner flies into the territorial airspace of any country without permission, then the air force would respond, not the civil air regulator.

So what does this have to do with international waters and maritime navigation? The air example shows how an equivalent fee for water passage would have to be implemented, when it concerns international waters. Everyone has to agree to the terms, the fee has to be economically reasonable, the contracting state must have some relationship to the area in question, and an international organization must actually contract with a qualified state to provide said service.

The present action in the Strait meets none of this criteria. So yeah, a ship could try their luck and pass through the Strait without making payment. Maritime and international law would not support the USA's claims for remuneration, even if adjudicated in an American court.