Is it unusual for a collaborator to ask for funding for a very short project?
Seems to me like that's their way of saying "no".  One week of work in exchange for authorship is a great deal -- and you'd give authorship to the professor, too, in exchange for nothing?  Why would the professor say no?  I'm guessing that the professor doesn't think your research will succeed, or get published, and therefore doesn't want to invest the time.

I'd recommend just working with the student in his "spare time".  I'm sure it's worth his time in exchange for authorship.