Well, I would believe that the first Jewish converts, in Jerusalem and environs, as well as the disciples, as recorded in the Gospels and Acts 1-9 or 10 or so and then in Acts 21:20 and on, were indeed the very first, first, first Messianic Jews, If they are not what would you call them? Obviously they had no NT in its completeness, if at all, other than probably oral carrying of the message. Therefore these Jews who had come to believe that Yeshua is The Messiah were "Messianic" in the ultimately original sense of the word "Messianic".
Therefore, it should be obvious, even to those that would deny it, that these orignal Messianic Jews were "Messianic" though they had no complete NT, but surely they were "Messianic". So,it follows that you can be "Messianic" without believing "every word of the NT". It's probably more important to BE Messianic than to worry about being considered this, that or whatever, by others.
HOWEVER, the New Testament, as a text, certainly is useful and informative to those learning of His Salvation and how to live His Way, and I think read and re-read to help continuing in The Way, rather than the mere man made machination of a Rome derived church(s) and its contrivances.
I would imagine that some may not like my response. Good!