Unicode supports adding a variation selector (characters U+FE00, U+FE01) to switch between "chancery" and "roundhand" script style:
https://en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols#Glyph_variants
https://www.unicode.org/charts/PDF/U2100.pdf (last page)
So LaTeXML could emit the appropriate selector to distinguish \mathcal from \mathscr. Other symbols support variation selectors, e.g. \emptyset and \varnothing can also be distinguished via the same mechanism.

Most fonts do not support the variation selector, and neither does MathJax, but variation selectors are supposed to be ignored, so there shouldn't be unintended side effects.

I'll submit a parallel issue to MathJax to hear if they can implement support for the Unicode selector in their MathML input.