I've kind of managed to do this. But I can't get it to do it so that when I send a response, it automatically changes to "Awaiting Customer Response".
I've created a custom status field, then I use triggers to change that status.
So I've set up trigger for when the ticket gets updated and the current status is now "Awaiting customer response". This then changes the current status to "Awaiting response" in a different colour to indicate it requires a response from us.
Problem here is when replying, I change the status to "Awaiting Customer Response" when updating the ticket, but the trigger changes it back and I have to change the status AGAIN as a private note.
So you kind of can, using triggers (and lots of filtering in them). But not nicely.