In your example, you wouldn't use the definite article anyway. "Eine Tänzer" could be wrong indefinite article or wrong noun, but unless you aren't hyper obsessed with gender (of people), it doesn't matter. "Die Tänzer" would only be used when they were introduced before. That's how define articles work. Sure, a foreigner could get that wrong, too, but then there are already 2 mistakes and of course, more mistakes make it worse. Also: if it's established that it's a dancing establishment, why not ask about "how many dancers" instead? As I said, you can construct cases, but it's not easy and yours isn't very good I'm afraid.

About your work environment: I get that it can come across as unprofessional, especially in writing. But in writing you have grammar checks, if you're lucky (I just checked and libreoffice sadly doesn't, but maybe with addons and I think MS office (which I'm forced to use at work) has it by default). And if they are viewed as morons in spoken form, I'm sorry to inform you that you work in a toxic environment. My non native colleges never had that problem as far as I know.