I have resolved this issue after an online chat with Support.

In my case it appears that it all stems from trying to send copy invoices to another email address, whether by ticking the 'Email me a copy at' box (Gear Icon/Account & Settings/Sales/Messages (Default email message sent with sales forms), or by entering an email address there in either the Cc or the Bcc box. Even ticking the 'Send me a copy' box in the Send dialog has the same effect.
I have given feedback about this so hopefully someone at QuickBooks will get this corrected sometime!