I suspect knowing the CVV isn't the issue. All the CVV does is confirm to the bank that you are the owner of the card. I have never heard of anybody having problems with a CVV. when removing a payment method unless it is actually when you are adding a new payment method which may be required you do first. To find out how to remove a payment method, and what to do if you can't remove your last payment method or use no payment method with your Apple ID, read this article ➔ Remove a payment method from your Apple ID - Apple Support
Unexpired subscriptions and Family Sharing organizers must always have a payment method present. To edit the cards used for those you may need to first add the new card and move it to the top payment method position before removing the old card (see: If you need to change or update your Apple ID payment method - Apple Support ).