One drawback of internet apologetics is the inability to use props. I can tell you what I would do right now, but I can't actually do it. But let's suppose I could. If you were here right now, I would offer you a £5 note. There is one condition: You can only take the money if you can prove it is real without referring to the note itself.
Money is a self-authenticating object. A real note can be distinguished from a fake note via circular reasoning. In this rare instance, this is not a fallacy. In fact, money is not the only thing that works this way. Other documents, such as licences or passports, also contain things intended to authenticate themselves and distinguish themselves from fake items.
"The Bible is true because it says it's true" is a fallacious argument. It is not one I endorse, and I don't think I've seen anyone outside of a satirical skit ever use it. However, there is absolutely no fallacy in reading the Bible in order to search for certain marks of authenticity. In fact, completely barring the Bible from entering the discussion is, itself, circular reasoning. It assumes the Bible's fraudulent nature in an effort to prove it. It would be like saying "no, you can't hold that bank note up to the light to see if it has a hologram."
So it's actually rather strange for atheists to have this phobia of a Bible verse. When discussing the authenticity of the Bible, it is actually very important to discuss what it says. We must use it to define our position, for example. One thing atheists love to say is that in order for us to believe in God, we need physical proof. In this case, it's not fallacious for a Christian to point out that this is the wrong kind of evidence to ask for, because the Bible says God is spirit (i.e. not physical). It is useful to compare the Bible's history to the world's history and show why the Bible's account is better. We can show that the Bible's authors are first hand witnesses to many of the events described therein. You see, then, how we can verify the Bible in much the same way as we can verify a banknote. It contains many unique self-authentication features.
The strongest of these is that its authors are witnesses to the death and resurrection of Jesus Christ. Human beings don't just rise from the dead, and so if Jesus rose, there's probably something to Him. And that's very important. The Bible says He died for a purpose. For you. You sin against God, but in His love, He doesn't want to punish you for it. Therefore, He punished Jesus instead. Jesus died in your stead, then rose from the grave, enabling you to repent of your sins, confess Him as Lord, believe He rose, and you will have eternal life because of it. There aren't enough self-authenticating banknotes in the world to buy what He bought for you with His own blood.