@AldritchM 
In my experience, a possible solution is to make sure there are no open invoices for the sub-company/client. Then, when editing the sub-client, in the sub-account/parent area, there's usually a check box that allows the sub to billed/invoiced with the parent or on it's own. You want this to bill/invoice NOT with the parent, nbut on it's own sub-account. I've been usccessful, on occasion, being able to make the switch, but sometimes not. If you can switch it to it's own, you casn then change or reassign it to another parent.
Probably only about a 35% success rate, but it's worth a shot to try before deleting/recreating invoices.
Also, in the process of deleting/recreating, it might be worth a look to see if when editing the invoice, you could duplicate the invoice and assign it to the new parent at that time. That would save you from having to manually re=enter all of the items, etc. and it would keep all of the exact billing cycles, etc. Haven't tried this myself, but it might be worth a shot.