All you care about is the mm part.

Programatically I would transform the date column in the array, to a 'season value' before doing any sorting.

Create a vector (assuming zero based) S with 12 elements s[0..2] == 0 (Spring),
s[3..5] == 1 (Summer) and so on.

For each row in your array, get the mm substring from the date. Convert the mm to an integer say i, replace the date value in the column with s[i]. Use a length(column) to see if the year 2000 problem exists.

If you sort numerically on the transformed date column, you get seasonal order. You can also preserve the date by adding another column to the array, transforming into it, then sorting.