The first problem with Messianic Judaism being “Biblical Judaism” is that by the use of this descriptor the clear implication is that Orthodox, Conservative, Reform and Reconstructionist Judaism are unbiblical and without basis in the Bible.
This may be the understanding of many adherents of this description. But the use of this title in a brush stroke discounts the development of Judaism over the last two thousand years as an unbiblical endeavor. This either demonstrates ignorance of other forms of Judaism or an attack on the larger Jewish world as being unbiblical, both are bad options.
The respect for the Bible, especially the Torah is so ingrained in Jewish thought and practice that it is beyond the pale to brand other Judaisms as unbiblical. Especially in living out the covenant relationship with G-d embodied in the Torah we have much to learn from other Judaisms, especially Orthodox and Conservative Judaism who in practicality live lives that are more Biblical than most Messianic Jews.
So then it can be seen that the use of “Biblical Judaism” is problematic in that it automatically brands all other Judaism as unbiblical which is clearly incorrect.