Date::Calc will do the job:

use Date::Calc qw/:all/;

my $days = Days_in_Month($year,$month)
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Check the docs for many more useful functions.

Update: Note to self, finish reading question before answering :)

You should take a look at the Day_of_Week() function, using that and the number of days in the month you can loop over the month and calculate the number of the days you're looking for. It isn't pretty, but it should do.