Avoiding Ceasar by BLV? Unlikely!

If you look at Bob Hale and Crispin Wright's Logicism in the twenty-first century, in this book (esp. pp. 168-169) (also: at Frege's works themselves and at other explications of Frege), the way things are supposed to have gone is this:
  • Frege used Hume's Principle (see my previous post) to derive second-order Peano Arithmetic.
  • Yet, he was unhappy with the explanatory role of Hume's Principle: "we can never—to take a crude example—decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or not."
  • So, he introduced extensions by means of Basic Law V (the extension of F is the same as the extension of G iff exactly the same objects are Fs and Gs), and defined numbers in terms of extensions.
While people usually think that Frege moved to BLV to avoid the Ceasar problem, I haven't found anywhere, neither in Frege, nor in secondary literature, a clear explanation of how BLV is supposed to help. Sure, once you have a theory of extensions in place, you have an explicit definition of what a number is. BUT, if you're Frege, you've introduced your extensions by means of an abstraction principle (okay, let's pretend for a while it's consistent with the rest of your theory), and this only moves your problem one level up:

We can never decide by means of our definitions whether any concept has the extension JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is an extension or not.

Am I missing something?