you are stating your customer would like to set up the SAP ECC Self-Billing process. You will find the proper process description and links to prerequisites in the SAP help file for this topic at http://help.sap.com/saphelp_erp60_sp/helpdata/en/1b/ca82d840fe11d393930000e8214d78/content.htm .
Note that Ariba CC processing is not the same thing as ECC self-billing: CC is the digital "carbon copy" representation in ECC of a paper invoice the supplier sends to the customer. Typically this CC process is used when the supplier does not have the technical capabilities to send invoices electronically through the network, i.e. when there is no ERP whatsover on the supplier side. The CC in the customer's ERP is sent back to the Ariba Network, so the supplier eventually has access to a digital representation of his invoice using his network account.
Self-billing on the other hand is a process that fulfills a different purpose: When the buyer consumes material sent as part of a delivery schedule, and (for example) material is prone to break, deteriorate, or change properties like e.g. weight or freshness during delivery, it would be impractical for the supplier to bill the customer for every single delivery and having to hash out the differences each time. Or, the customer may just be in a stronger position, and mandate that he will only pay for material he sees fit for production at his discretion, and discard the rest at the supplier's risk. Hence he is able to mandate he'd write the invoice on behalf of the supplier but based on his actual use of the material, and at intervals agreed upon with the supplier, typically in conjunction with scheduled payment runs. The supplier would typically have tolerances configured in his ERP system. When comparing open values of his internal invoice with the received self-billing document values the supplier's system automatically posts all differences, debit or credit, depending on the +/- sign. It assigns a reference number to all documents that are assigned to the delivery. The reference number is used later to post receipt of payment. If the supplier had set a tolerance check, the system creates no open items unless the tolerance levels are exceeded. If the tolerance levels are exceeded, the system creates new open items. However, these are not assigned a reference number. Unless tolerance checking is set, the system creates new open items for every clearing entry.
For setting up electronic automated communication between supplier and buyer, please check this SAP help document on EDI Settings for the SD Self-Billing Procedure (as Ariba's cXML won't help you when there is no Ariba Network involved): http://help.sap.com/saphelp_erp60_sp/helpdata/en/2a/dac353b677b44ce10000000a174cb4/content.htm?frameset=/en/1b/ca82d840fe11d393930000e8214d78/frameset.htm¤t_toc=/en/0e/8db6535fe6b74ce10000000a174cb4/plain.htm&node_id=19&show_children=false
When it comes to Invoices, from Ariba Network (AN) it is always Outbound and the invoice generated in AN are sent to Ariba Downstream or nowadays also to SAP ECC. Hence I am a bit confused when you say "And PI delivers it directly to the recipient; instead of posting it to the ASN".
Having said that, I would request you to please update this thread on how you managed to close this as it would be good to know for many.
Can understand your confusion. Still looking for all the right terms and so on.
In short; in the SAP document it is mentioned that you can post your data directly from ECC to ASN. Another option is to do this through PI. As the client saw this they wondered if it was possible to send the data to PI. And instead of then sending it to ASN would it be an option to sned the cxml document directly to the supplier.
But as Andreas mentioned "Ariba's cXML won't help you when there is no Ariba Network involved - See more at: http://exchange.ariba.com/communications#sthash.y6gIJEuh.dpuf", I think we need to conclude that the client's idea won't work.