I don't understand the need for the term "Messianic Jews". If they are Christians, then they should be glad to be called by that name.
Mashiach, Messiah, and Christos, Christ, are simply two translations with the same meaning, which is "Annointed One." Mashiach is the original, since the promises of the Messiah came in Hebrew, not Greek. That doesn't mean that it's wrong to say Christ--the Apostles certainly didn't mind translating the word for their Greek audience--any more than it would be wrong to say "Jesus the Annointed" to an English audience for the sake of explaining the meaning of "Christ" and avoiding our tendency to think of it as a surname.
The reason that Messianic Jews choose to use "Messianic" instead of "Christian" is twofold: First, because they are not ashamed of their Jewishness, and wish to incorporate it into their worship and self-identification.
Second, because there are in fact distinctions between a Messianic perspective and a mainline Christian perspective. I can tell you for a fact that many Christian churches receive a Jewish believer into their ranks and immediately put the pressure on him to eat a ham sandwhich and celebrate Easter instead of Passover and put off all things Jewish as a rite of passage.
There's a reason why the Jewish community considers "Jewish Christian" to be an oxymoron. For more historical perspective on that reason, you might want to read this article.
By using the term "Messianic," a congregation emphasizes its Jewish character and its commitment that Jewish believers in a Jewish Messiah should be able to stay Jewish and keep the Torah that Yeshua did instead of being forced into Gentilism.
There's something of a debate right now as to whether those of a Gentile pedigree, like myself, should use the term "Messianic Jew," "Messianic Hebrew," or whathaveyou. (Sadly, many congregations, fearful of losing their Jewish character, exclude Gentile believers in clear contradiction to Scripture. I suppose that's one way to avoid having to deal with the terminology issue.) I personally just call myself a "Messianic" to avoid being seen as making a claim to something I'm not.
I've taken to preferring saying "Yeshua" to "Jesus" not because of any sacred-name nonsense (the idea that if you are saved in the name of Jesus instead of Yeshua, you're not really saved), but to emphasize the Messiah's Jewishness, and because Yeshua, in Hebrew, means "salvation" (the longer form, Y'hoshua, means "YH(VH) is Salvation"), and I like to preserve that. I often use the Hebrew names of the Apostles to emphasize that they were all Jewish and Torah-observant as well.
In any case, someone calling themselves "Messianic" or "Messianic Jewish" should no more bother you than someone calling themself a Baptist, Calvinist, Lutheran, or whatever. Just consider it a denominational name, and don't put up a wall of separation over it.Shalom.