You can do it in a fairly straight forward way using Non capturing groupings combined with {} for Matching repetitions:

#!/usr/bin/perl
use strict;
use warnings;

my $string = 'Gyatso is a bad monk as bad as bad bad as bad bad bad ba
+d bad.';
my $n = 5;
print ($string =~ /(^Gyatso(?:.+?bad){$n})/);
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