Search committees that I know read all the letters written for the candidates in whom they are interested, though of course some people read some letters more closely than others. I will admit that at my "research-focused" university, I often read teaching letters very briefly. In my defense, many teaching letters are not worthy of more than a very brief reading. They simply do not separate out the candidate or provide any really incisive or useful information. If teaching letters are to be believed, then approximately 99.5% of all candidates are above average teachers. Why do I find myself skeptical of this?
That said, I don't really see much advantage to "hiding" research in a teaching letter (or vice versa). If you get a teaching letter written by a "research professor", then they may be a little nervous about not mentioning the research whatsoever or feel through force of habit that they must say something. But really, when we read a teaching letter we want to hear about teaching, so saying something at the beginning like "Dr. Ray has asked me to write about her teaching, and so I will" would be fine.
You speak of a "bounded number of letters". That must be true, but the supremum may not be what you think. In most jobs that I know of, nothing stops you from submitting more letters than the number they ask for. Past a certain point maybe it looks obnoxious: I remember one candidate for an assistant professor position that had nine letters of recommendation (8 research, 1 teaching, I think). I felt that was a bit much but we certainly didn't disqualify the guy or write back to say how many of the letters we had burned without reading. In fact he made the short list and almost got an interview.
I would say: if you want to send one or two more letters than the application asks for: just do it. If you want someone to talk about your research, don't have them do it in a way that other people might not read, or might read but dismiss because they are expecting to hear about the teaching. Just get another letter. For that matter, it is increasingly common to get more than one teaching letter even for a research job, even though I know of no research job that asks for more than one.