when you flush it not about the temperature, you could be in the snow mid winter, you still flush,
fans really help, because the air moving over your skin, helps your body re regulate, with a fan you overcome the flush far faster than without, room temperature has nothing to do with the flushing,
A fan cannot interfere with the air con, they have made that up as an excuse, it is really to do with aesthetics , tell them to buy a dyson fan. Silent and unobtrusive, suggest it as a compromise.
I can get quite panicky when I am out of range of a fan when a major flush hits, it's very difficult to function, fans make you recover far quicker, so huge sympathies, it would be very unreasonable to not allow you a fan.
only someone who has never experienced full on flushing would object, anyone who has to deal with it, would understand it totally necessary.