I think you're way over complicating this, if a cable or interconnect is time shifting the signal, it will show up in a null test of two recordings, one from the "suspect" and one from a reference. Software like Audio Diff Maker will time sync and level match automatically, before running the null test, ie: reverse phase on one recording by 180 degree and subtract that from the other recording. All you need is a good sound card to make the recordings, which you need for REW anyway.
Sure you don't get waterfall plots or impulse response graphs, both of which require interpretation, but you'll get an inkling of any timing errors by examining the difference signal in comparison to the original in any wave editor and/or by listening. What you will get is a number for the null difference and can decide whether that's humanly possible to hear in the presence of music.
Trying to discern differences with a $100 mic over speakers and REW is IMO, going to be really difficult and/or misleading, too many degrees of separation from measurement and DUT, but probably the way I'd go if I wanted to "prove" cables/interconnects make a difference.