Found this interesting write up about how the NBA should find a way to get Sixth men involved in the All-Star game and I was on the fence when I started reading but I agree with it. It sort of makes the case for the 6th man being involved similar to how relief pitchers and closers are selected to the MLB All-Star game.
I think it would need to be an additional spot, let the selection process work as is but add a separate spot for 6th man voting. That can be divided equally amongst fan voting, player, press, and coaching selections as well.
Having 13 All-Stars per team shouldn't be an issue. There were 13 each last year when Silver added Dirk and Wade to the All-Star game based on their career accomplishments.
https://sircharlesincharge.com/2020/01/19/nba-all-star-game-vote-sixth-man/