You could implement paypal direct payment (I think it costs $30-$40 per month). Then you could take all of the credit card information from your website and not go to paypal. That would include the billing information so you would have it. As long as you support paypal payments, you couldn't get access to the billing information I don't think.
I did see a few places where it mentioned REQBILLINGADDRESS but I don't see it in the paypal documentation. You could try contacting paypal to see if it can be enabled and I could help you integrate it into your paypal request.