Might I suggest a re-think of what is meant by "wears the name of Christ"?
It has been strong in our Church of Christ culture to think in physical terms, of having the label "of Christ" on our buildings and our web sites and our bulletin letterheads.
But I'm reminded of the second of the Ten Commandments: You shall not take the name of YHWH your God in vain. For years, I took this to mean that we weren't supposed to *use* the "name" in an improper way, such as saying, "Oh, God, this pizza is good!"
The Jews took it that way too, which eventually evolved into their thinking that the name shouldn't be used at all, which is why they then altered the written scriptures to hide the name of YHWH behind the generic term "Lord" (the way most of our printed English Bibles still do).
But since then I've come to believe that it's more like a bride being encouraged to honor her husband's name by living honorably with him; don't take your husband's name in vain by having dalliances with the milkman and the pool-boy; be faithful to the name you have taken.
If you're going to wear the name "of Christ", then behave worthy of the name.
I'm not sure that "wearing the name of Christ" really refers to having "Church of Christ" on the sign out front of your meeting place; I think it refers to having "of Christ" in your everyday behavior.