Here is the AT&T approved way to perform what is called an "Upgrade Swap" on a family plan.
It's really fast, easy, approved by AT&T and does _NOT_ require the swapping of sim cards.
Why make this any harder, when it's actually quite easy
Step 1) Order the phone on the line that is eligible for the upgrade.
Step 2) Upon receiving the phone, simply call AT&T tell them you are doing an "Upgrade Swap".
They will move the phone numbers to the proper phones without any further time wasted, and activate your new iPhone for you.
Now you are all set.
This is precisely why I always order through AT&T since the phones come with new sim cards installed, and you end up with both a new phone and a new sim card. Apple cannot make these simple changes to a family plan.
The entire process takes approx six minutes.