There are four problems with your code. First the $_ in the grep() is assigned to an arrayref, you need to dereference the ref to get at the first element of the inner array of your AoA:

my $GrepResult = grep($_->[0] =~ /$SearchStr/,@cdata);
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second you are executing the grep() in a scalar context .. in that context it will return the number of items matched, not the item(s) themselves:

my ($GrepResult) = grep($_->[0] =~ /$SearchStr/,@cdata);
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third, the regexp would match even strings as 'not_the_test2.txt.gz':

my ($GrepResult) = grep($_->[0] =~ /^$SearchStr$/,@cdata);
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that is you want (I believe) to find only the exact matches not the first file that contains the search string.
Which leads to the last problem, the $SearchStr might contain some special characters ... actually it already does, the dot. The dot means "any character" within a regexp. So you would need

my ($GrepResult) = grep($_->[0] =~ /^\Q$SearchStr\E$/,@cdata);
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to make sure such characters are escaped. In that case, it's easier though to get rid of the regexp completely:

my ($GrepResult) = grep($_->[0] eq $SearchStr,@cdata);
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If you did need a regexp and were looking for something not exact there is one more thing to notice. qr//. If you do

my ($GrepResult) = grep($_->[0] =~ /^$SearchStr$/,@cdata);
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then the regexp is compiled again for each item in @data. This could be expensive, especially if $SearchStr was really a regexp and was complex. In that case it's good to do something like

my $regexp = qr/^$SearchStr$/;
my ($GrepResult) = grep($_->[0] =~ $regexp,@cdata);
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This way the regexp is compiled just once and the search is quicker.