Can one flatmate do cleaning instead of paying toward a shared cleaner?

We had agreed that the bathroom needed more regular attention. Then the cleaner's quote arrived: $120 for one monthly clean of the shared kitchen and bathroom. Split three ways, that would be $40 each. Kai said he couldn't add another monthly bill and offered to clean the bathroom himself.

I do most of the shared cleaning now. The third flatmate wants the service because a booked visit feels more reliable than our old rota. Kai says the old rota failed because the tasks were vague and everybody swapped weeks, not because he refused to help. That is a fair description of what happened, including the weeks I changed my own turn.

The quote is for both rooms. We haven't asked whether removing the bathroom would change the price, and we haven't defined what Kai would do or when. If the quote stayed at $120, the other two of us would pay $60 each while he did his part as work.

Would that be a reasonable arrangement, or should we make a clearer household rota before adding a service? I want relief from chasing chores as well as relief from scrubbing.