D is still a year away from applications, but we are starting to slowly gather information she’ll need.
How awkward is it going to be for her to ask her private music instructor for a letter of recommendation when he is the professor at her first-choice school? 
She loves this school, but unfortunately, it is very expensive. So we have to have a Plan B in case the hoped-for scholarship isn’t enough to make it manageable. (this school allows for academic scholarships, but they have a cap… which is a tiny fraction of the COA of the school. Anything else from the school itself has to be talent-based… and, of course, we have no crystal ball for that information! 
I’ve read that asking shouldn’t be awkward, but it always seems framed in the case where the teacher/professor is at a lesser-choice school vs first-choice school, so she and I are both curious - should she approach him in a very casual, “of course he’s expecting this request” kind of way? Or should she further explain why she’s applying to other schools at all?
I would imagine that the school being cost-prohibitive will organically come up in their conversations and lessons over the coming year - so maybe the reasoning will be obvious to him when the time comes?
Every college professor will assume and expect any student to apply to more than one school and, if he’s her private teacher, it will be his role to help her identify potential schools, including finding some financial safeties. Many college faculty know the faculty at other schools, so you really should take advantage of his knowledge in this.
And as her private teacher, he will also be fully expecting to write a letter of recommendation so, yes, very much in the “of course he’s expecting this request.”
And your daughter is going to need to discuss pre-screen/audition requirements with him, so she’s going to have to discuss the actual list of schools with him.
I think it makes it easier that he teaches at her top choice. Schools like to hear that they’re the top choice, because they can give the best offer they can. Offers tie up scholarship money; they don’t like using it for people that are not likely to accept it. I think you and she should be honest with him about financial constraints; faculty often have a lot of say (sometimes all the say) in how merit awards are distributed.
For what it’s worth, my son’s private teacher was from our local University. There was no “conflict of interest” problems during college applications. When decisions came out, they had several pow-wows to help my son make his decision and the teacher was really supportive. My son often said that he wished it had been his top choice because it would just have made the whole process so much easier.
Not awkward at all. It is not uncommon for students to study with conservatory or college profs including the “top of he field” institutions. I know high schoolers (including my own kids) who had recs from profs at places like Juilliard, Indiana, Northwestern, etc. Many teachers write letters for their high school students knowing they will be sent off to other schools. As per ScreenName, your D’s teacher should be a big help in recommending colleagues and schools to her. I suppose there might be a handful of teachers whose personality is such that they expect to keep kids forever, but that would be unusual and problematic IMHO. Is it unusual for your D’s teacher to take high schoolers? If not, the instructor has probably been through this multiple times before with other students.
Some teachers actively encourage students to go elsewhere for college and study with new teachers. We had this experience in our family and I know others who have also
Thank you! I am relieved. 