That's really interesting. You use a TeX editor for lilypond, presumably for syntax highlighting and checking? I had some vague idea lilypond was loosely based on TeX in the original days, but I did not think it is compatible now. I though the '\' notation was the only surviving archaeological DNA fragment.
I use vim with point and click in Ubuntu. When you get it all up and running and add snippet packages and so on, it is a pleasure to use. As a long time emacs user, I find the present emacs lilypond mode is very buggy and effectively unusable, for me anyway, and messes up indenting quite badly. A pity. Although I have used emacs forever, I have never spent time coding emacs, so it's not really my area to fix it. With a little love from some kind developer, that would also e a very good environment.
On your point re compiling - Frescobaldi is not compiling, it's lilypond. So that is a curious thing. Perhaps you lilypond command settings in Frescobaldi are different to what you set up in TeXShop.