I'm not sure if I have more to offer than what folks have said here, but I'll afflict you with my opinion anyway.
Beyond sharing a common sense of purpose to bless the lives of the needy, I'm not sure I am concerned with being included in the community of Christians. We
are different, and in a doctrinal sense I'd say we we even
want to be different.
Where being known as Christians, in addition to Mormons, is helpful is in conveying our values to people who don't know much about us. We want folks to know we believe in Christ "and him crucified" as Paul would say. We want them to know we believe in being "honest, true, chaste, benevolent, virtuous, and in doing good to all men." Outside of the obvious missionary benefits, it disarms people when they feel they share something in common with you.
The problem of being persecuted for not being Christian is fortunately growing smaller, in my opinion, and I speak as someone raised in the south. I don't mean to demean others' negative experiences, but I always felt that whatever ridicule I received could be converted into something positive if handled properly. I also find that given our history, Mormons sometimes have a persecution or inferiority complex, and may take offense more easily than is necessary.
My 2 cents.