There is no such thing as a "Wallet ID" when it comes to the bitcoin protocol. Any centralized service which provides you with a "Wallet ID" is simply using it to process logins to their website or system. It is not recorded anywhere on the blockchain. Further, knowing a single address from one wallet does not allow you to calculate any of the other addresses within that wallet. You might be able to work out (with some assumptions) other connected addresses by looking at different addresses being used as inputs together in the same transaction, but if someone is careful not to link addresses in this way, then you might not be able to tell anything at all.

I would never recommend using an exchange as a surrogate mixer. Unlike a good mixer, all centralized exchanges track your IP address, your deposits and withdrawals, perform blockchain analysis to see where your coins come from and where they go, and link all that information together to sell or share with third parties. Using an exchange might prevent your family member from identifying your wallet, but it gives you much worse privacy overall, not to mention the additional risks you accept of having your coins frozen or seized.