but it’s mashma from the megillah that he died on Purim.
No, it’s not. In fact, the megillah makes it fairly clear that he did not die on Purim.
Haman cast his lots on the 13th of Nissan.
Esther fasted for three days (13-15 Nissan) and on the third day had her first meeting with Achashverosh and Haman.
The second meeting (where Haman was sentenced to die) took place on the next day.
Esther 8 makes it clear that some time passed between Haman’s death and the day that the king’s decree against the Jews was to be enacted (which was 13 Adar).
The Wolf