As the cap says 1922 1923 it may suggest a sport with a season that covers both years?
The badge is the arms of King Edward VI. Could it be part of the King Edward's foundation schools, possibly the High School?I agree but if you look many of thefootball and rugby caps are very similar in style especially the 1920 ones. I still think it might be worth contacting the colleges and seeing if they can help or point in the right direction to find out.
You are quite correct and the majority of pupils had the red badge, my self included, however for those few "super sportsmen" the colourful alternative was not only a badge of pride but an incentive to others to be the rare recipient of such an honour.The only King Edward's badge I vaguely know is Camp Hill Boys which is all red.