MD Longhorn wroteThe BR is out the moment it is declared fair. The ball (assuming it's over fair territory) is fair the moment of the interference. She is not a retired runner at the moment of interference.
(PS --- if she IS retired, then the out she supposedly prevented by interfering with the fielder catching the fly ball has already been recorded - you can't call the same person out twice.)
So, I disagree with that initial premise; the batter is NOT out when it is declared fair, the batter is out when she hits a ball that can be judged an IFF in an IFF situation. The ball is assumed fair until it is not; and THEN, if determined not fair, the batter is not out. That is why we declare the IFF even if unsure it will be fair; we can always reverse that status after the fact with less jeopardy. And, thus, the batter is out, even if not declared initially; because she hit a ball that can be (and should be, and therefore IS) an IFF.
This is similar to our instruction that every pitch is potentially a strike; and we should consider it a strike, until it isn't.
These basic premises allow us to see "border line" situations as possible strikes and outs; it helps us maintain the edge to see the outs (and strikes), wherever and when ever they occur. It may be easy to see balls and safes whenever there is a close play; but that isn't why we are there.