44. Al Khazari: It is strange that you should possess authentic chronology of the creation of the world.
45. The Rabbi: Surely we reckon according to it, and there is no difference between the Jews of Khazar and Ethiopia in this respect.
46. Al Khazari: What date do you consider it at present?
47. The Rabbi: Four thousand and nine hundred years. The details can be demonstrated from the lives of Adam, Seth and Enōsh to Noah; then Shem and Eber to Abraham; then Isaac and Jacob to Moses. All of them represented the essence and purity of Adam on account of their intimacy with God. Each of them had children only to be compared to them outwardly, but not really like them, and, therefore, without direct union with the divine influence. The chronology was established through the medium of those sainted persons who were only single individuals, and not a crowd, until Jacob begat the Twelve Tribes, who were all under this divine influence. Thus the divine element reached a multitude of persons who carried the records further. The chronology of those who lived before these has been handed down to us by Moses (Kuzari 1:44-47, 1905 translation by Hartwig Hirschfeld)
The discussion is about the "creation of the world" which would imply the six days of creation preceding and inclusive of the creation (or giving of a soul to) Adam HaRishon. I think this is further emphasized by "the Rabbi"'s response in 1:61 that reliable information that the world was older than 4900 years would challenge his faith, which at very least would be inconsistent with a view that tool wielding hominids roamed the world prior to one of them being given a neshamah.
Does this settle the matter, of course not. But I do think that the Kuzari can be included among those who see no reason for “אֵין מִקְרָא יוֹצֵא מִידֵי פְּשׁוּטוֹ” to inherently exclude the beginning of Bereishis.