The seventy-two-fold name is regularly presented as ancient, and the impression is usually created by leaving the dates out. They are worth putting in, because this is the claim about the system a knowledgeable reader will check first.
The Talmud discusses divine names of unusual length. At Kiddushin 71a it speaks of a name of twelve letters and a name of forty-two letters, with rules about who might be taught them. It does not describe a name of seventy-two letters, and it does not describe the Exodus 14 derivation.
The earliest clear attestation of the technique is Rashi, the northern French commentator who died in 1105. Commenting on Sukkah 45a, on the phrase ani va-ho, he writes that these belong among the seventy-two names which are marked out in the three adjacent verses in the portion Beshallach, and he then names them: va-yissa, va-yavo bein machaneh, and va-yet Mosheh et yado. Those are Exodus 14:19, 14:20 and 14:21. The method is set out, in a standard commentary, at the end of the eleventh century.
The Bahir, a foundational Kabbalistic text usually placed in the second half of the twelfth century, works with the same material, and the Zohar, from the late thirteenth, elaborates it at Beshallach 52a. The Sefer Raziel ha-Malakh tradition carries it further.
So the sequence is: a Talmudic interest in long divine names, without this one; an eleventh-century commentator who describes this one as already known; and a medieval Kabbalistic literature that develops it.
That is a respectable pedigree of roughly nine hundred years. It is not a biblical one, and the system should not be allowed to borrow authority it does not have.