If the said Airbus was designed to chime when a dispatcher sends a message (I read in previous posts that it does not), maybe we wouldn't even be having this conversation right now.

ATC calls our operations daily requesting that a particular flight tunes to a particular frequency. We (as dispatchers) have the option to "flash" or "chime" the ACARS message. In this case, I would have chose to "Chime" my flight crew. We also have the option to require an acknowledgment. Again, in this case I would require my flight crew to "acknowledge" that they did indeed receive my ACARS message.

If it's a message from crew sked, or an ACP asking a bout a delay, I simply would send the message with no chime.

We have A300's, B747-400's, MD-11's, B757 and B767's. I believe flight control has the ability to chime these ACARS at any time, even below 10000 feet. (Please save that discussion for another thread)