If the command runs quickly:

my $out = `cmd-to-run 2>&1`;
# Should really check command return code via $? special variable here
# Then print $out via your $tee here
# Note: 2>&1 (to fold stderr into stdout) works fine on Windows XP
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If it is a long-running command and you want feedback while it is running:

open(my $fh, 'cmd-to-run 2>&1 |') or die "open error: $!";
while (<$fh>) {
    # print $_ via your $tee here
}
close($fh) or die "close error: $!";
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And autoflushing stdout ($| = 1;) is prudent in these type of scripts to avoid any intermingling of command stdout and your script's stdout.