First the string is parsed to an internal javascript string representation, so you first need to escape for javascript string literals. The result of this would be the same as what you already have - so no modification is needed.
Now you have
<div class = "markup4"><br>blablablabla1.<br><br></div><div class = "markup3"><br>blablabla2.<br><br></div>
Internally in javascript memory.
After this, it's just like you would write that as normal HTML, so of course, if you use "
it will not work.
It can get pretty complicated to store strings that go through 2 different interpretations, which is why you shouldn't do this but use templating engines with the templates stored in other files or custom script elements on the page, which would allow you to focus only on the html interpretation.
" != '
– Mitzvah