Try installing a handler for SIGPIPE in the child. You do make the call, but don't die if it failed.

our $PARENT_ALIVE = 1;
$SIG{PIPE} = sub { $PARENT_ALIVE = 0 };

[...]

$unix_sock->print($message) if $PARENT_ALIVE;
if (! $PARENT_ALIVE) {
    print "note: parent died; will not attempt future messages.\n";
}
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It's been a while since I actually did this kind of thing, though, and this is untested!

(You can also subclass IO::Socket::UNIX and override print to wrap this safety check for you, if you want.)