If you look at his Example 1 (C major to G major), you'll see that he doesn't include the F♯ in the figures because it's assumed in G major. The same is true in Example 2, the F♯ and C♯ are assumed in D major, so there's no need to put them in the figures.
When he gets to Example 6, though, he needs to throw in those accidentals on the Neapolitan sixth, because, since he's resolving to F♯ major, he needs to clarify B♮ and D♮. (You've already addressed these two issues in your original question, I'm just setting the stage.)
The double-sharp first appears in Example 8, and the only possible explanation is that it's a courtesy accidental. Looking through the book, he always uses a courtesy accidental whenever there is a double-sharp; the example in your question is just one of many.
I've been looking for a citation to prove my answer, or maybe scores of his where he also follows this rule, but I've been unable to find one. But looking throughout the book, it must be that Reger only uses double-sharps as courtesy accidentals in the figures.