It is not really about what they use to clean that it is only a power challenge.
Having been T.U.P.E."d they want to show you who has the power, they will push back to most things.........it is your business so they will do it your way, showing how much better you are by doing half a floor against them doing the other half is only pandering to their whim.
Unless you want to show how better you are in all areas of work stand up, direct what you want how you want and even why, make sure they know who is boss. Staff always think they are the important ones that they are the indispensable ones THEY ARE NOT, YOU ARE
By the way you will not find better cleaning products than Tadghs, no really I mean it, having tried most of the usual suspects Tadgh is our choice
cheers
Gordon