File handles can only be used to transmit bytes. The warnings indicate you are trying to transmit characters that aren't bytes, meaning something that doesn't match /^[\x00-\xFF]*\z/.

You need to convert the text you are trying to send into bytes. The process is a special case of serialisation known as character encoding.

You may use Encode's encode, or you may add en encoding layer to the file handle.

open(my $fh, '>', ...) or die ...;
my $buf = encode('UTF-8', $text);
sysrwite($fh, $buf);  # Or: print $fh $buf;
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open(my $fh, '>:encoding(UTF-8)', ...) or die ...; 
sysrwite($fh, $text);  # Or: print $fh $text;
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binmode($fh, ':encoding(UTF-8)');
sysrwite($fh, $text);  # Or: print $fh $text;
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is there some minimally-invasive way to wrap [my $image = chr(0xff) . pack('cN', $someThing, $otherThing);] so that when I syswrite it, I get good old-fashioned binary data out?

That code always produces bytes, even for invalid inputs. The resulting string will never cause "Wide character" warnings.