"I Am That I Am" in the Gospel of John?

God Appears to Moses in Burning Bush (1848)

Eugène Pluchart

Saint Isaac's Cathedral, Saint Petersburg

(Image from Wikipedia)

About a week ago, I posted a hermeneutic on John 17:11b, speculating on the possibility that the verse should be read as meaning that the Son bears the Father's name, i.e., "Yahweh":

Πάτερ ἅγιε, τήρησον αὐτοὺς ἐν τῷ ὀνόματί σου ᾧ δέδωκάς μοι, ἵνα ὦσιν ἓν καθὼς ἡμεῖς.

Holy Father, keep them in Your Name, which You have given me, that they may be one as We are.

My suggestion was that the clause "Your Name, which You have given me" implies that the Father has given His Name to the Son. Since that post one week ago, my Uncle Cran has reminded me that John 8:58 rather clearly identifies Jesus with Yahweh:

εἶπεν αὐτοῖς ὁ Ἰησοῦς Ἀμὴν ἀμὴν λέγω ὑμῖν πρὶν Ἀβραὰμ γενέσθαι ἐγὼ εἰμί

Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I Am.

As Uncle Cran notes, this alludes to Exodus 3:13-14 of the Burning Bush theophany, especially to vere 14, where God reveals his name to Moses in a pun on the Tetragrammaton "Yahweh" (יְהוָה):

וַיֹּאמֶר אֱלֹהִים אֶל־מֹשֶׁה אֶֽהְיֶה אֲשֶׁר אֶֽהְיֶה וַיֹּאמֶר כֹּה תֹאמַר לִבְנֵי יִשְׂרָאֵל אֶֽהְיֶה שְׁלָחַנִי אֲלֵיכֶֽם׃

And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.

There's some debate over how to translate the Hebrew, e.g., "I Am That I Am" or "I Will Be What I Will Be," but the allusion in John's Gospel is fairly clear, as the Septuagint might also suggest:

καὶ εἶπεν ὁ θεὸς πρὸς Μωυσῆν ἐγώ εἰμι ὁ ὤν καὶ εἶπεν οὕτως ἐρεῖς τοῖς υἱοῖς Ισραηλ ὁ ὢν ἀπέσταλκέν με πρὸς ὑμᾶς

The red-fonted expressions are "I Am The Being One" and "The Being One," respectively. While a more rigorous argument would need to be constructed, the evidence -- cf. ἐγώ εἰμι (i.e., "I Am") in John 8:58 and in Exodus 3:14 of the Septuagint -- seems to me to support my suggestion that John 17:11b, in stating "Your Name, which You have given me," implies that the Fourth Evangelist presented the Son as bearing the name of the Father, i.e., "Yahweh."

But I've not yet constructed a rigorous argument, which would be time consuming . . .