It's a little weird, really, for every team to have Sox in their name somehow. No matter what team sponsored them, it's odd. "What team do you play for, Timmy?" "White Sox Yellow." It's not THAT big of a deal, but it is strange.
Of course, when I was in Little League, our teams had MLB names and we wore that team's cap, but our jerseys had the name of our sponsor on the front. (I was on the Pirates but my jersey said Zemsky Bros.)
I think some sort of White Sox patch on the shoulder would have been a decent compromise, along with one team obviously being the White Sox.