Your redirection is in the wrong order:

my $command = "echo $writeline >> $file";
my $output = `$command 2>&1`;
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Writing that out, you end up with

my $output = `$command echo $writeline >> $file 2>&1`;
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Which means redirect STDOUT to the file, then STDERR to (current) STDOUT, i.e. also to the file. You probably want

my $output = `$command echo $writeline 2>&1 >> $file`;
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instead, i.e. redirect STDERR to STDOUT, then STDOUT (but not the redirected STDERR) to the file.