It makes more sense to assume that the two views in the Midrash are not in disagreement about the calendar. Since there was no mitzvah of kiddush hachodesh yet, the determination of how many months in the year must have been according to some cheshbon, or maybe according to G-d, but it’s not the sort of thing that easily lends itself to disagreement about in sevara. It’s fair to assume everyone agrees that the year Moshe was born was a leap year – the question is whether he was born in Adar 1 (so the three month count ends at Nisan) or in Adar 2 (and it ended in Sivan).
Is there any way to figure out whether the year Moshe died was a leap year or not? I have no idea, but maybe someone else does.