A simpler solution may be to use an en dash, even though they are not interchangeable and em dashes are the proper punctuation for parenthetical phrases. As a typography pedant, I’m annoyed that LLMs have forced us to talk about this.

I think this is more of a style issue than one of correctness: lots of high-quality typeset output has used em dashes for parenthetical phrasing and plenty has used (spaced) en dashes. Bringhurst is a partisan for the en dash, for example, saying that "The em dash is the nineteenth-century standard, still prescribed in many editorial style books, but the em dash is too long for the best text faces." (/Elements/ version 2.5, p.80).

Of course, if we collectively shifted to the spaced en dash then LLMs would eventually follow; it's not clear to me that any simple and deliberate sign of humanity could remain exclusive given the incentives for machines to replicate it.

Modern British style tends to prefer spaced en dashes over tight-set em dashes for parenthetical phrases.