bochur1818. in my opinion ,first of all , the waY to approach this is gradually. start spending a bit more quaLITY TIME WITH YOUR FATHER, LEARN WITH HIM IF POSSIBLE, IF YOU DON’T ALREADY, AND LET HIM SEE THAT YOU ARE A GOOD ,SINCERE PERSON. WORK ON YOUR COMMUNICATION AND RELATIONSHIP SO HE FEELS RESPECTED, VALUED LOVED AND HONORED.He may feel threatened that his way isn’t good enough for you ,or he may believe that “A Hat does not make The man” so show him that you honor and repect him and that’s the first step. Step two, Tell over stories of Gedolim, and what you want to emulate in their behavior, at the shabbos table, as part of a dvar Torah, or quote a dvar Torah from a Godol you want to emulate.. There are many sources for such stories. Step three, tell your father you would like to wear a hat because it makes you feel a connection to these Gedolim, and be sure to mention Gedolim he admires and considers as his Rebeeim. If you do this gradually you’ll be preparing the head which deserves to wear a hat.
eventually , your father will hopefully realize that this is for you and won’t stand in your way.
We had a similar situation when my son wanted to wear a certain type of Yarmulka when he started high school. I was always against labeling and divisiveness and on purpose refused to let people tell me what specific headgear was supposed to represent. I always said a man wearing something on his head is supposed to show that he believes and remembers that Hashem is Above him. I’m not interested in knowing anything else. But of course we live in another world , not my fantasy world , so when my son started High School he asked us if he could switch yarmulkas. i asked him Why. He said because i stand out, i’m the only one in this yarmulka not that one. I answered him that it is a valid reason, not wanting to be odd man out and so he switched yarmulkas. the next time he came home from yeshiva, he had been in the house for several hours before someone else came in and commented on the new Yarmulka. I had never even noticed it, because to me he was my son and always will be no matter what yarmulka or hat he wears. ( P.s. it took me years to get used to the beard,(where did my cute little boy go
?) now hE is 34 Bli ayin Hara and I’m finally used to it after 10 years) good luck to you, it may take some time for your father to come around