"What do you mean with that" can be grammatical (in standard English) if it is a question about quoted speech:
What do you mean, "with that"? With what?
What do you mean with that?
is avoided by speakers. A Google n-gram comparison pf "do you mean with that" versus "do you mean by that" gives some evidence that "do you mean with that" is very rare. Furthermore, it doesn't mean anything different and can be replaced by "what do you mean by that".
It's rare enough that those who don't use it and haven't heard it will regard it as wrong or as some regional dialect, and if a non-native speaker uses it, it will be seen as a non-native mistake.
Note also that some of the matches for "do you mean with that" include sentences having forms such as "Which Mary are you going with? Do you mean, with that girl from across the street?" which is not what we're interested in. An "n-gram" is just a string of words.