Contrary to what you are being told here, using a pseudonym is neither weird, nor uncommon. Actors, musicians, and other performers do it all the time and we never think twice about it. It may seem odd to some to do it in the context of a dance class, but I can think of plenty of good reasons to do it, among them all the ones you mentioned. Privacy is not an uncommon concern these days, and for very good reasons. I use a pseudonym for songwriting, and MOST of us use them right here on the forum, and all over the Internet.
Nothing at all wrong with it.
What you have to keep in mind though, is that it's almost impossible to keep people from figuring out what your given birth name is forever. Sooner or later, some will figure it out. If all you do is remain aware of that, and be prepared to explain and be honest about why, and cop to who you really are once people do find out, there's nothing at all wrong with it.
I have a cousin who did just that, and not for performance reasons. He just didn't like the name he was known by and opted to start telling people it was something else. To all of us, he is still Scott. To everyone that met him as an adult, his name is Jim. None of his friends has a problem with it. His wife of over thirty years has no problem with it, and never did. Those who are close to him know, and those who aren't don't. It's never been a problem for him.
You are you, no matter what you choose to use as a name. It's if you lie about who you are that it's not just weird, but is basically conning everyone.
"A rose by any other name..."
It's just a name.
Too late now for this class, as I see you already chose not to, but if you ever think about doing it some time in the future, well, why not?
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I'm not likely to be around much longer. As before when I first signed up here years ago, I'm finding that after a long hiatus, and after only a few days back on here, I'm spending way too much time here again already. So I'm requesting my account be locked, banned or whatever. It's just time. Until then, well, I dunno...