What you are looking for is not a Bitcoin ATM machine, but how to hide yourself. For example, to verify that you are the owner of that coins, you must search for the address with any block explorer, which means that your IP address can be collected.

+ It is wrong to trust the private key that was printed by the machine, they may keep a copy of it and therefore you and that ATM (or anyone who has access to its database) owns that money even though you paid for it.

Your best solution is to send those coins to a new address and learn to increase your privacy with Bitcoin.