I would do a little eye dialect when the character first speaks:
Thash nuyor biznush!
(Maybe not quite that dramatic, but maybe.)
Then have your other characters react to it, asking the character "sorry? I didn't quite get that?" or echoing what they think they heard. Then, because all of this is tiring for the reader, switch from dialog to narration:
Eventually Perry was able to establish that Chris had seen the rider yesterday, and heard his voice this morning.
You could also try not specifying the precise words:
Chris mumbled a quick denial, almost unintelligible but definitely angry.
You can also summarize with dialog amongst others:
Any luck talking to that old drunk up there?
Chris isn't drunk, it's some sort of speech impediment. Anyway, she told me …
If the character re-appears after many chapters, again go with the eye dialect to remind everyone that Chris mumbles, and then again switch to just summarizing. You have lots of ways to avoid literal dialog from this character.