I want an option to "Notify Affiliate Every Order", but NOT just cc the notification email. That contains the purchasers information, and I don't want my affiliates to see that.

You should have an option to simply send them an email saying something like:

"Congratulations, You have been credited with a sale of the product xxxxxxxxxx. You now have accumulated nnn sales and $nn.nn in total. The ballance we owe you $nn.nn and we will pay you once your balance reaches $nn.nn."