This will be difficult to do. Depending on your backend, Plots uses Mathjax or a homebrew version thereof to render TeX strings (it doesn’t actually use the fact that your string is a LaTeXString, but rather the fact that LaTeXStrings include $$ automatically, so this is a question for Plots rather than LaTeXStrings).
For the Plotly and PlotlyJS backends, you can use a user-defined config file:
https://docs.juliaplots.org/latest/backends/#MathJax
which I guess could be used for this. but I don’t know MathJax, I saw someone say that the available fonts are limited anyway.
If this was an issue for me I would probably use the PGFPlots backend, so you’re actually leaving the rendering to LaTeX itself. If the compilation times get too large, you can add a recipe to your makefile to compile those image files to pdf before including them in the document.