How would the scarlet items effect work in the following scenario?

Reddish is correct. You'll get a 3-item Superior Gollux set. The shoulder will do nothing (unless you also have Empress items on, or any other set that includes a Shoulder). Joker items only count as their own type: hat can only stand in for hat, shoulder can only stand in for shoulder, etc.

Also, there is no sweetwater accessories set, so there's no question of any scarlet item trying to get into that set.