I think your best bet is to change the LaTeX version. Some thoughts that suggest this solution:
Using MathTeXEngine.jl:
julia-1.10> generate_tex_elements(L"\mathcal{J}")
1-element Vector{Any}:
(TeXChar '𝓙' [index 2819 in NewComputerModern Math - Regular],
[0.0, 0.0], 1.0)
(Ignore the terminal font… :))
I understand that MathTeXEngine.jl always uses the NewComputerModern(Math) fonts, and here uses the Unicode glyph U+1d4a5, “MATHEMATICAL SCRIPT CAPITAL J”. The index 2819 here is simply saying that the Unicode glyph U+1d4a5 occurs at index 2819 in the font file.
However, the pdfLaTeX example you show looks to be using the letter J from the ComputerModernSymbol font:
and there doesn’t look like there’s any Unicode going on here.
The last time I used LaTeX I used LuaLaTeX, which someone said was the newer replacement for pdfLaTex…
Presumably LaTeX allows you more flexibility in selecting and using fonts… There’s a few fonts with Unicode Script Js - although to me they often look more like other letters rather than J. Your best choice looks like Cambria Math. 
