I don’t’know how your DJ controller and mixer work together. If all of “final sound” sound comes-out the mixer you can record from its USB connection.
A lot of people seem to have trouble recording DJ controllers but I believe the Serato software has recording capability.
Any of those should work with the mixer. MME is the oldest protocol and WASAPI is the newest.
If the “final sound” is coming out of your soundcard you can record it with WASAPI loopback. The other two don’t support “loopback” but sometimes they have “What-U-Hear” or “Stereo Mix” which also record whatever is coming out of the soundcard.