In short, the way I see it, your options are:
1) Actually make affiliates issue an invoice before getting paid. As seen from DDF's example, this is very hard, and you'll probably miss out on many potential affiliates, if you're just starting up now.
2) Explore the legality of self-billing invoices.
https://gfy.com/fucking-around-and-pr...payouts.html#7
This is not really a common practice in the biz currently. And you still might have trouble getting some affiliate's real names' and addresses to put on the self-billing invoices I believe.
3) Setup another company in a country that doesn't require invoices
4) Have an outside biller handle the affiliate payouts, eg. CCBill. I think they simply pay you for the income, minus their fees, minus affiliate commissions.
PS. People I've talk to treat Paxum just like a bank account in the accounting.