I'm not entirely sure it makes a whole lot of sense that Mike Barnes would return in Cobra Kai.

In The Karate Kid Part III (where Mike Barnes is from), there wasn't even much of a rivalry between him and Daniel LaRusso. The only reason they even fought each other is because Barnes was a publicly known karate champion at the time, that Terry Silver hired with his wealth. Prior to Silver hiring Barnes, they were basically strangers who didn't even know each other. So, in addition to the non-existent rivalry between Barnes and LaRusso, it wasn't like there was even a compelling student/mentor relationship between Barnes and Silver, either.

In the end, Barnes was just some stranger that Terry hired to get back at LaRusso in the tournament, and what ultimately happened is that Barnes lost.

With all these things in consideration, there's not much of a reason to think that Silver would ever recruit Barnes again, or feel the need to reach out to him on any sort of personal level.