The oath doesn't contradict the mitzvah. The mitzvah is to live in Eretz Yisroel, and the oath says that we must live without sovereignty, under another nation. Therefore, one can fulfill both by living in Eretz Yisroel under a non-Jewish government. However, if the non-Jewish government ruling the land is not accepting any more Jewish immigrants, and the only way to live there is to overthrow the government or conquer the land, then it is forbidden under the oath.

In many mitzvos we find that there are permitted and forbidden ways of doing the mitzvah. For example, if you cannot get to shul on Shabbos without driving a car, then you must stay home. If you cannot afford tefillin without stealing, then do not put on tefillin. This is not called a prohibition to fulfill a mitzvah; it just happens that in that person's case he cannot do the mitzvah permissibly. In other cases it is permitted.

The same is true here: there are cases when it is permitted to live in Eretz Yisroel during exile and fulfill the mitzvah, so the oath is not coming to directly contradict the mitzvah.