Hi,
If you don't get the billing information after the Lead is converted I'm not sure how you can avoid entering the information again in some way. However, very little information is needed to create a lead. What I'm thinking is that when creating the lead you can create a second contact that is the Lead contact and hold that information. You can then base all of the lead related activities on that second lead. Then, when they convert to a customer, you will just fill in the billing information for the main contact.
Method mimics QuickBooks in that the main contact is created based on the information entered when the customer is created. Therefore, when a Lead is created, the main contact is based off of the Lead. Based on your process, even if you just used QuickBooks and took Method out of the equation, you would still need to update the information when they are billed.
To simplify this, another thought would be to customize a button that would create a new Contact based on the main contact. This way, entering the existing contact only requires a button click instead of re-entering all of the information again. You can then just update the main contact with the billing info.
Hope this helps.
Adam