OP's bottom line question is whether he will succeed in a chargeback.

The only way I see that happening is if the shower is not addressed by either repairing it or moving OP to another room AND OP leaves and moves to another property. In that case, if his card is issued in a jurisdiction which permits partial chargebacks, he may succeed if the property insists on charging him for the time after he leaves. But, this is by no means certain and thus, OP should only do it if he is willing to risk having to pay for both rooms.

If OP does leave, he should have a clear record of the date, time name & title of each person with whom he spoke and the specific request he made. He should also have photos and documentation of the specific shower defect (which really needs to make the room unsanitary not merely a nuisance).