With regard to the specific example in the question--the Gemara Megilla 3a says that
כהנים בעבודתן ולוים בדוכנן וישראל במעמדן כולן מבטלין עבודתן ובאין
לשמוע מקרא מגילה מכאן סמכו של בית רבי שמבטלין תלמוד תורה ובאין לשמוע
מקרא מגילה קל וחומר מעבודה
Priests at their [Temple] service, Levites
on their platform, lay Israelites at their station — all desist from
their service in order to come and hear the reading of the Megillah.
It was in reliance on this dictum that the members of the house of
Rabbi were wont to desist from the study of the Torah in order to come
and hear the reading of the Megillah. They argued a fortiori from the
case of the [Temple] service.
There is a dispute between Ran and Taz how to understand the Gemara. According to Ran (among others) this does not mean that if one must choose between the two, one reads the Megillah, but rather that one reads the Megillah first, and then performs the other mitzvah. If one could perform only one, it would be the avodah, and not Megillah. However, Taz (OC 687) disagrees and maintains that the Megillah has priority even when one can only perform one mitzvah. Even according to Taz, this is a special rule regarding Megillah, but it does not apply to other mitzvot derabbanan. Taz explains that Megillah has the status of a Biblical mitzvah because it is mi-divrei kabbalah--prophetically sanctioned--and not an ordinary rabbinic mitzvah. Thus, we see that in the case of an ordinary rabbinic mitzvah, one would not prioritize it over a biblical mitzvah.
Seemingly, then, the Mishnah in Avot does not mean that one can prioritize rabbinic mitzvot over biblical ones. Rambam in his commentary on the Mishnah cites the rule of העוסק במצוה פטור מן המצוה, one who is involved in a mitzvah is exempt from another mitzvah, as a corollary of the principle of this Mishnah. With regard to העוסק במצוה, R. Elchanan Wasserman (Kovetz Shiurim 2:32) is unsure whether this applies to a mitzvah derabbanan as well. Even if we assume it does, though, this would not mean that one can give the rabbinic mitzvah priority when one is not yet involved in it.
(Chida, in his commentary Z'roa Yemin on the Mishnah suggests the following interpretation as well: הוי זהיר במצוה קלה--be scrupulous about a rabbinic mitzvah, such as the restriction about blowing shofar on Rosh Hashanah that falls on Shabbat, כבחמורה, as for a biblical mitzvah such as shofar itself; והוי מחשב שכר מצוה כנגד הפסדה, and keep in mind that the risk of violating the rabbinic command may be greater than the reward of fulfilling the biblical mitzvah...)