In my opinion, toolbar buttons would be best (or a dropdown box with "symbols"). This would be a site-specific section in the toolbar area at the right of the standard buttons. There are probably other sites within StackExchange as well that have similar requests for special symbols (for example Mathematics).
It would also be nice to have the double sharp symbol, which also has a unicode, but not many fonts support it: 𝄪 𝄪
If this should be used, it means that the font used here on SE music must have support for this.