Hi Timo,
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.
Regards
Vilas