(Life has evolved on the edge of a knife, at the narrow balance point between enormous energies that cancel out just so. I often think that at the beach, looking out across the ocean, marveling that the water is almost never sloshing around at any scale proportional to itself. It's up to us to educate those who don't understand positive feedback loops and the existential risk they present to any system in equilibrium.)
You mean watt days (watts * days).
It is akin to thinking that "2 apples" and "an apple divided by 2" are interchangeable because both expressions involve the concept of an apple and the number 2.
watts/unit thus seems fine to me, whatever the unit may be, even if it itself is derived from time. watts per day would just work out to joules/second/1/24*60*60, making 1 watts per day a derived unit that expresses joules/84600 seconds, or an instantaneous rate of one 84600th of a joule.