You said all feedback so...
2) Occasionally an invoice covers multiple orders.
I see that as a major part of the problem. Multiple invoices per order sure, but not one invoice for multiple orders. No wonder things sometimes don't match (#3). IMO there should only ever be a 1 to 1 or many to 1 (invoice to order) relationship but I'm no accountant.
To allocate AND status order parts I think you could solve that with order line item records. Then you could assign the line item to a cost center and status it as well as edit the status or reallocate portions of the cost. However, if you need to split a line item among cost centers you could not split a line item with a straight 1 to 1 line items table, you'd need a 1 to many design. However, I only got that you want to split orders, not anything like line items.
This all reminds me of a blanket order release system I worked in. One "order" tied to one PO which is tied to one cost center and has a dollar ceiling. Each release is like a new number (a release is like a line item on a PO). I thought it was a great system but it wouldn't support your current business model.
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.