Paying for groceries with privacy is not only possible, it's extremely common. That's the entire concept of a membership card (at stores which will sell to the general public).
I can use the store and not flash my membership card if I don't want them tracking a particular purchase.
This option doesn't exist with Facebook, you can't opt in and out from click to click. And they share the information with any number of outside parties.
Furthermore the charge that Facebook is levying vastly exceeds the value of the information they're forgoing if I pay them.
You might have more trouble than that. Did you use a payment card for the secret purchase? Have you ever used the same card while providing your membership information?