Just connect the mixer to the computer with USB. There’s no reason to have another device connected to get audio from the mixer to the computer… and it’s probably to a detriment with a little thing like that, anyway.
While you can only use ASIO usually with one app at a time (sometimes I can pull two off with an MP2015), you can have both ASIO & Windows Audio running at the same time on the mixer. So you could send music from VDJ 7 to the mixer in ASIO and/or record back to VDJ 7 or just ASIO into Soundforge when running with SC5000s and/or be streaming to OBS or Wirecast at the same time in Windows Audio. I do it all the time.
By the way, be careful about how the sound is set up in streaming apps. Make sure the outputs you want on the mixer are actually selected properly and every other audio connection, microphone, desktop audio feed, etc, is either deleted as an option or muted in the app. Very easy to have a bunch of other stuff running with sound that you don’t realize until later.