Would cooking meat and milk in the microwave be prohibited on the biblical level or just a rabbinical level prohibition?
Consult your local Orthodox rabbi.
In terms of cooking on Shabbat, there seems to be one understanding that it is merely a shinui (e.g. Rav Moshe Feinstein); and another understanding that it simply is not considered bishul (Rav Shlomo Zalman Aurbach), which makes for derabanans in certain cases. This distinction is made in regards to the korban Pesach, matza, challah, etcetera. Maybe we can extrapolate from this and extend it to cooking milk and meat, according to the latter opinion? A close examination of the sources would be required.