Hi,
I don't know of a solution for that.
The Shipping invoice is only available via the button "shipping invoice" in the backend when you edit an order.
That would require developing a plugin to send such email with the shipping invoice generated as a PDF and attached to the email before sending it.
So it would require some development.
We have actually developed a plugin to send a PDF of the invoice to the customer after the order is confirmed:
www.hikashop.com/component/hikashop/product/cid-18.html
So if you would base your development on that, it could greatly reduce the amount of development necessary, but it will still be more than 1 or 2 lines to change.