Updating the customer on a ticket does not trigger an ticket update in trigger terms.
This means you can’t create a trigger that “resends” those information. In fact, technically this could get in the way for “normal” tickets.
I suggest to ensure that your agents do not forward mails, but redirect them.
This will keep the original mail headers in tact allowing Zammad to recognize the original sender of the message.
This also spares you from changing the customer of the ticket.
Alternatively you can use subfolders Zammad fetches from to move mails from mailbox to mailbox if that’s better. This is especially better for Outlook users, because “Reply-To” and “redirect” functionality is a really huge pain within Microsoft world.
Another alternative would to not use triggers for “we received your request” stuff, but a scheduler.
However, this adds a delay for every ticket up to 10 minutes and requires you to ensure the condition no longer affects “already notified” tickets on the next scheduler run.
If you don’t break the condition for a scheduled ticket, you’ll end in spamming your users up to every 10 minutes on the same ticket. That’s something you (and your customer) potentially don’t want.