the fwupdate command may reset the remote machine without doing a proper shutdown, leaving TCP connections unclosed.

Instead of closing the channel, try closing the socket and then disconnecting the full session:

my $socket = $ssh->socket;
my $chan = $ssh->channel();
$chan->exec("/sbin/fwupdate -m\n");
close $socket;
$ssh->disconnect;
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Though, even then there is a race condition: fwupdate may be faster taking control of the machine than the SSH server responding to the SSH_MSG_CHANNEL_REQUEST request.

Setting blocking to 0 could help there, but them, the exec method may fail and you will never know if the command was delivered or not:

my $chan = $ssh->channel();
$ssh->blocking(0);
$chan->exec("/sbin/fwupdate -m\n");
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Alternatively, if you are running your script in a Unix/Linux box, instead of Net::SSH2, you could use Net::OpenSSH (the OpenSSH client used by the module under the hood is able to detect staled connections):

use Net::OpenSSH;

my $ssh = Net::OpenSSH->new($host, timeout => 30);
$ssh->scp_put($firmware[$antenna_type],'/tmp/fwupdate.bin');
$ssh->system("/sbin/fwupdate -m\n");
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