How can I calculate what a recording would sound like if it was played through my speakers and mic? I am trying to isolate the sound of someone speaking while music plays back in a noisy environment. 

I have multiple speakers and a microphone in my system. I am able to 


play back just music and record it, then 
play back music while speaking and record it. 


After that, I can look at the difference in the signals and get the meaningful information I need.

In order for this to work in real-time, I would need to calculate what that first recording would sound like. If I have the impulse+magnitude response or some other data about the system, is it possible to predict what the microphone will detect?

EDIT:I suspect that some kind of transform could be derived from the impulse response that will give a decent prediction of what the mic will hear. To be clear, this doesn't need to be incredibly accurate. I just want a general idea of what the mic will pick up.
You can use convolution to hear a recording as it would sound through a specific speaker. The linked page describes how to get an impulse response and convolve a signal with it. 

https://www.mathworks.com/help/audio/examples/measure-impulse-response-of-an-audio-system.html