I think Scrivener’s ability to embed any sort of file can get you half way there.
It is true that alien file types can’t live in the Draft folder, but as a songwriter, why would you be living in the Draft folder anyway? For projects which house many independent items, like songs, poems, short stories, it is better to live in the Research folder, and just drag folders up if needed into the Draft folder for compiling. Often enough with such projects, Compiling is largely irrelevant.
You can create an audio file with nothing in it and bring that into the Template folder in a Scrivener project. Once you’ve done this, you can use the Project > New from template item to create blank sound files anywhere in the binder (except the Draft folder, of course).
For example, I maintain a project for a writing group and each of the sub-projects is a short story which needs to be typeset eventually in inDesign. For which purpose, I have an otherwise empty InDesign doc in the Templates folder of that project. So, each subproject gets carried up to Draft for Compiling, but then I create an inDesign doc for that story inside Scrivener and in the folder of that sub-project (back in the Research folder). That doc opens in inDesign, of course, and the story gets imported and processed to produce typeset output, but the end result is that all phases of these story projects are housed within Scrivener.
I imagine something like this process could be usefully pursued in your case.
So, you would not, in the end, be hitting the record button inside Scriv, but would be hitting it in some chosen helper application (I use Sound Studio for a waveform editor, but also am not a musician). So, not quite what you asked for, but given that the sound files would be housed in Scriv and can be launched from there, it is mighty close — perhaps on due consideration as close as you could really want.