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 w
As you noted, the letter was most likely sent in early August. Lily's choice of the word "still" makes it sound as though 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.