For your interest, I worked on exactly this emailed invoice problem over many months. Our solution ended up having several layers.
Firstly, the invoices need to go through APR one at a time (I see you already concluded that).
We then put an email check in the APR flow, so obvious problems get diverted to AR staff for checking.
We then bcc’d an internal address so there’s a totally non-technical record of what was successfully sent.
Then there was a UD flag against the invoice header which was set to true if and only if the SysTask tables had record of successful completion, with a daily report of invoices still false after posting.
It was a lot of work over a long time, but seems to be (touch wood) failsafe so far. Obviously you can’t prove the customer received the invoice, but then you can’t prove a hard copy in the post got there either. Statements and a good AR department catch the oddities as long as you minimise all the other problems.