I agree with you, but unlike Electricity Retailers, Water Utilities regard the property owner as their Customer, primarily because the old charging regime for water and sewerage charges was based on land value (like Council Rates) – not water used or sewage discharged.
As a tenant, you’re only liable for the water usage component of the Account, but I don’t see any way of avoiding disconnection of the supply other than as you suggest, to pay the entire Account and make an adjustment to your next rent payment. Just make sure you keep records, and advise your current Agent in writing / e-mail about what you’re proposing to do; that is before you do it!