Hey folks - just posting a resolution to this one. Stanislav found the issue.
It turns out that you should only pass the is_support parameter when you want the ticket to be created as a Support ticket. The absence of this parameter implies that you're creating an usual ticket. I resolved the error I was getting by omitting this parameter when it was false (i.e. don't include it in the JSON or XML when it's false).
Ideally, the API should probably be returning an HTTP-400 in this case - not an HTTP-404, which was confusing - but at least there's a simple solution.
They are going to take care of the documentation.