if (grep(/^$check/,$_)) { is a rather interesting line. In essence it is the equivelent of if (/^$check/) { because there is only one element in the list for grep to work on ($_).

That aside, you need to strip the .zip off before the test, not during the test. One way (there are many) would be:

my $check = 'test.zip';
(my $test = $check) =~ s/\.zip$//; # strip the zip
$_ = 'test';
print "match" if /^$test/; # Note no grep - no need
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