Ok , I can see your point, but I bet you do not dress down to a greasy tee shirt from Lew's garage and a pair of cut off jeans either.
The hat with your wings on it,that's earned appropriateness for any occassion, especially in these times. You could wear it to dinner with your tux and it wouldn't bother me.I hope you wear it everywhere with pride.
With my tux, I wear a mismatched pair of cuff links. One is from USMA and one from USNA.The first belonged to my older brother, the second to a friend. Both did not make it home from Viet Nam.I don't care if they don't match.
By the way, how people dress is part of how they act IMHO