Yes, it is reasonable to write a separate teaching and research letter, for exactly the reasons you write above: application readers often separate the teaching and research letters in their minds and (at research institutions) they may not give much attention to the former letter. (Although if the teaching letter is written by a sufficiently famous mathematician who is an obvious leader in the applicant's field, I think the danger of this is much reduced.) Note also that recommendation letter inflation being what it is, many sharp-eyed readers will snap at almost anything as a sign of lack of strength. Talking about the candidate's teaching when you are supposed to be (in the eyes of the reader) talking about their research will definitely be taken by many readers as a sign of research weakness.
On the other hand, if I ask for four letters -- three research and one teaching -- and your application gives me four letters, but two written by the same person, it is natural for me to feel a little cheated. Surely part of my request was to get opinions from four different people?
So here's what I will recommend: if one person has serious things to say about the person's research and their teaching, s/he should write separate letters (and briefly explain this in both of the letters). However, at the same time one more letter should be included in the application: i.e., all together, one teaching letter from Professor A, one research letter from Professor A, and three more research letters from Professors X, Y and Z. Note that one of X,Y,Z should probably be the student's thesis advisor (in particular, the advisor should not be Professor A).
I have never seen this arrangement in an application, but I can find nothing objectionable about it. Indeed it would look to me like the candidate had taken the issue very seriously (and, of course, followed my advice, for which many bonus points would be awarded (-: ).