Cancelling out known signal with sampling offset
What you are describing is basically an acoustic echo cancellator, as used in e.g. any car with a «hands free» phone solution. Use the (known) dry song track as a reference, and use a stochastic gradient search algorithm (eg NLMS) to online identify the FIR filter which relate your recorded signal to the known reference. As long as the channel (room acoustics) remains relatively stationary (LTI) and the ambient noise is uncorrelated to the reverberated music, you should be able to adopt standard methods to your application.