The actual quote from Clement is as follows:
And that it is said, that we and the Greeks know the same God, though not in the same way, he will infer thus: "Neither worship as the Jews; for they, thinking that they only know God, do not know Him, adoring as they do angels and archangels, the month and the moon. And if the moon be not visible, they do not hold the Sabbath, which is called the first; nor do they hold the new moon, nor the feast of unleavened bread, nor the feast, nor the great day."
Clement is saying that neither Christians nor Greeks worship in the same manner as the Jews. Clement is not referring to weekly sabbaths at all, nor associating weekly sabbaths with lunar cycles. The reference to the 'sabbath' as part of the expression "the Sabbath, which is called the first" is in reference to the 'feast of trumpets' (Rosh Hashanah) marking the 'Sabbath' (that is, rest) commanded on the first day of the (civil) new year. See also Leviticus 23:24.
It would be redundant to say the first day of the seventh month is also a sabbath if the first day of every month were a sabbath anyway.