Filter away sinusoidal noise properly
Definitely, try to reconstruct the sine and then subtract it.  A notch filter will create artifacts as the desired signal changes, with notes stopping and starting, and percussion. Even if those notes are at pitches of frequencies outside the range of the notch filter.

To extract the unwanted sine, use a bandpass filter. While this has the same problem as the notch filter, with notes stopping and starting, you do know that the sine is steady (or slowly varying) over time, while the desired signal is always changing.  Or may include some steadily droning elements - bagpipes, anyone?   Hope that's outside the passband of the filter.  You also have found how the sine and the signal related between left and right, so make use of that to help extract the sine.  The electronics engineer part of my brain wants to send the extracted sine to a PLL, to filter out any accidentally included signal, and provide smooth amplitude and phase data. Some sort of mathematical flywheel to steady the cycles.

The exact methods for extracting and purifying the sine depend on if you need to do this in real time, or near real time with a fraction of a second delay acceptable, or if you have the whole span of signal from beginning to end all at once such as with an .mp3 file.

I know this is the best way to do it, because Cassini's NAC camera had a similar problem. You have the audio version of what I had to fix years ago.