I'm not aware of any other way to apply for an emergency travel document - they don't advertise this.
I should have provided this link that is, I think, at least partly relevant:
https://www.canada.ca/en/immigration-refugees-citizenship/services/canadian-passports/travel-documents-non-canadians/apply-how.html
Basically it says "contact the embassy or consulate" if you're abroad and therefore need emergency assistance for travel docs.
So there is a way to apply but no details provided. The chicken-or-egg issue is I'd guess they also want to establish the applicant is a PR and not inadmissible, which is usually done via the prtd process. (Plus there may be concerns/issues of whether the applicant has travelled to the home country from which protected person status was ostensibly needed, another relevant matter).
Anyway the point is well taken: the order of operations, how one applies for what in what order (or all at once or something else) may matter a great deal indeed. And crucial to avoid tripping the wrong refusal at the wrong time.
So a very specific case in which I for one don't wish to make any guesses. I think the point about the OP being a minor without documents (and this and related concerns being a reason for not applying earlier) is important, but there's a lot in here that's important, including procedural.