I have this issue as well and I did a BCC to my email and I did get it but my customer didn't receive anything. On the other hand, I was successful only once with a different customer. Shopify is there any resolution to this yet?
I've found a workaround. When you are creating the invoice and about to send it, you are shown a preview of the email. Scroll down into the email to the big red button thatnsay COMPLETE YOUR PURCHASE, and right click on it. There you can copy the link and send that to your customer. I like to verify the link in a new tab first. Hope that helps.