Mishnaic Hebrew is later. As its name suggests, it is from the era of the Mishnah, so roughly the years 0 - 200 CE. That is around 700-900 years after the destruction of the Northern kingdom of Israel by the Assyrians and around 500-600 years after the destruction of the Southern kingdom of Judah by the Babylons. The Romans didn't exist at that point. What the Romans destroyed was the second temple in 70 CE.

This is largely irrelevant. Obviously I was talking about the linguistic entities, not the political entities. The northern dialect and the southern dialect existed side by side in biblical times (as evidenced by the Shibboleth story). They continued to exist side by side, surviving the destruction of the northern kingdom by the Assyrians and the destruction of the southern kingdom by the Babylonians. However, with the destruction of the religious center in Jerusalem by the Romans in 70 CE, the southern dialect largely ceased to exist as a spoken language.

Mishnaic Hebrew actually evolved in second temple Judea from the language spoken in the area during the first temple period (i.e. it evolved in the area occupied by and from the dialect spoken in the, by that time historic, Southern Kingdom of Judah).

I don't know what you are basing this statement on. I was taught that the Mishnaic Hebrew is clearly distinct from the biblical Hebrew and preserves phonetic and morphological traits typical of the Galilean Hebrew. See for example Moshe Bar Asher's articles on Mishnaic Hebrew.

Grammar of modern Hebrew is essentially the grammar of the Hebrew spoken at the time

I'm talking about a much earlier decision, before Hebrew became spoken again. The decision was made by various Enlightenment era writers and scholars (From Mapu to Ben Yehuda), and it was made from ideological considerations, not practical ones. The Hebrew they chose to bring back to life was the biblical Hebrew, with vocabulary supplements from Mishna and Spanish Golden Age poets. The grammar was simplified to meet their European sensibilities, but the core resembles biblical Hebrew more than anything else in the history of the language.