What you have is a buffering problem. You have to turn autoflushing on for the new STDOUT file handle you make:

use IO::Handle;  # up top

...
    open STDOUT, '>&', $WRITE or die "failed redirect to STDOUT: $!";
    STDOUT->autoflush(1);
...
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Or you can use the perlvar $|, aka $OUTPUT_AUTOFLUSH

use English qw(-no_match_vars);

...
    open STDOUT, '>&', $WRITE or die "failed redirect to STDOUT: $!";
    $OUTPUT_AUTOFLUSH = 1;
...
[download]

I almost mentioned the select builtin but it gave different output, interestingly!