You can use mmm-mode available at GNU Elpa.
There are examples how mmm-mode is to be configured.
In your case the following Elisp lines in your init file should work.
When you open a LaTeX file with markdown environments those are highlighted in grey.
If you put point within those regions markdown-mode becomes active (e.g., indentation).
When you write a ...
The answer to my problem was a comment from The_Blackbeard user at reddit.
Have you taken into account that while emacs-lisp-mode derives from prog-mode, tex-mode derives from text-mode?
Once I had that information the rest was easy to fix.