I’ve been using wordometer for word count. It does character count as well.
There are some things it can’t count properly. Look at some open issues on GitHub.
A follow up question: what is your use case? Do you need the count in your rendered document, or outside just as a metric?
I think at the moment the more accurate way to go about a whole document character count will be to use an external script after rendering a PDF. So, have a toggle to disable some things you don’t want counted (page numbers, etc), compile it, then convert the PDF to text, do a count.