numbers up to 8 are sent, nothing after that

That implies that your UART (or UART emulation) has a hard limit buffer of 15 -- strange in this day and age, but not impossible -- in which case, perhaps it will like it better if you only send a max of 15-bytes at a time.

Try adding your own write method wrapper that breaks the string up into 15-byte chunks. Eg.

sub myWrite {
    my( $port, $data ) = @_;
    for my $chunk ( unpack '(a15)*', $data ) {
        $port->write( $chunk );
        sleep 0.1; ## may not be necessary; or maybe 0.001 is enough; 
+experiment.
    }
}

...
# $ob->write( "$string\n");
myWrite( $ob, $string );
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Of course, if you have time to waste, you could subclass Device::Serial and override write ...