I only just found post, but I'll share my thoughts anyway. I noticed the same thing during my latest reread of the series.

It wouldn't surprise me if JKR simply didn't notice, even though the contradicting text passages are within the same book. The books are generally very vague on when the Potters learnt they were in danger and when went into hiding. But even when I think JKR made a mistake, I try to find a convincing explanation to get rid of inconsistencies like this.

As you noted, the letter was most likely sent in early August. Lily's choice of the word "still" suggests that Dumbledore had borrowed the cloak a significant amount of time before this. It also sounds as though Sirius already knew that Dumbledore had borrowed the cloak. Lily or James had probably told him about it in a previous letter. Considering all this, Dumbledore must have borrowed the cloak months before the Potters'death.

I don't think Dumbledore confused months with days, no matter that he was old. I deem it most likely that he lied. Even in the King's Cross chapter he's still a bit