I have found another way that works, it can be temperamental and a pain at times, but it does eventually work. 

In situations where an invoice has been created outside of the project, I go to the estimate and first move that over to the project.  It didn't pop up with a message about unlinking transactions like it did with the invoice. 

Now that the estimate is in the project, go to the corresponding invoice.  Under the customer name, tell it to unlink.  This will remove all the line items on the invoice, but it will open the side menu that will allow us to link a transaction.   

Find the estimate that the invoice was created from (which should now be in the project) and relink it to the invoice.  This will repopulate the invoice with the data from the estimate.

Now you can go to the customer name and freely change it to the Customer:Project without having to delete/copy the invoice.

In situations where the estimate has already been paid, go to the transactions tab in the customers file.  Click on the payment entry.  This will open a new window where you can apply the payment(s) directly to the invoices, which you should now see listed as being in the project.

This has taken me the better part of a week to get the this process down. Like I said, it doesn't always want to cooperate, particularly in relinking the estimate to the invoice.  All day yesterday, it didn't matter what I did, it would not add information from ANY estimate once I unlinked it.  I gave it a go first thing this morning and it went like clockwork, took about five minutes to put two invoices into two separate projects and reapply the payments.

I hope this helps!