Can't email a partially paid invoice
When emailing invoices, the statuses the contribution can have are hard-coded to "Completed", "Pending", and "Refunded". However, you can still print invoices with any contribution status, and the invoice template supports partially paid contributions.
I'm going to take the minimally invasive route here, but I'm not sure why this is limited at all. It looks like this work was done in 4.6 by Rohan Katkar/WebAccess - not sure who to talk to about this, I assume this work was done on contract for another shop?