If either argument doesn't specify a directory, the current directory is assumed.

use File::Spec::Functions qw( catfile );
use File::Basename        qw( dirname );

my $src_qn = $x->{filename};
my $dir_qn = dirname($src_qn);
my $dst_fn = "TEST$ext";
my $dst_qn = catfile($dir_qn, $dst_fn);

rename($src_qn, $dst_qn)
   or die("...: $!\n");
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My naming convention:
fn = unqualified file name
qn = qualified file name (i.e. rel or abs path)
fqn = fully qualified file name (i.e. abs path)
And only pass qn/fqn to system functions.

Some of the lines can be combined, but I wanted to be explicit for clarity and demonstration purposes.