Would you be open to this statement if "begotten" were understood to indicate the way in which the Son relates eternally to the Father?
No. The Theological Dictionary of the New Testament has a lengthy article on the Greek word translated "only begotten". It has an explanation of the word itself (see below) near the beginning of the article and a paragraph near the end (see below) that supports the Creed's wording and your understanding of it.
Explanation of the Word "μονογενὴς" ("only begotten")
The word does not occur in Homer but is attested from the time of Hesiod. In compounds like διο-γενής, γη-γενής, εὐ-γενής, συγ-γενής the -γενής suggests derivation (γένος) rather than birth. Nouns as the first part of the compound give the source, e.g., from Zeus, the earth. Adverbs describe the nature of the derivation, e.g., noble or common. μονο-γενής is to be explained along the lines of εὐγενής rather than διο-γενής. The μονο- does not denote the source but the nature of derivation. Hence μονογενής means “of sole descent,” i.e., without brothers or sisters. This gives us the sense of only-begotten. The ref. is to the only child of one’s parents, primarily in relation to them. μονογενής is stronger than μόνος, for it denotes that they have never had more than this child. But the word can also be used more generally without ref. to derivation in the sense of “unique,” “unparalleled,” “incomparable,” though one should not confuse the refs. to class or species5 and to manner. [Büchsel, F. (1964–). μονογενής. In G. Kittel, G. W. Bromiley, & G. Friedrich (Eds.), Theological dictionary of the New Testament (electronic ed., Vol. 4, pp. 737–738). Eerdmans.]
Paragraph in Support of the Nicene Creed
What Jn. means by ὁ μονογενὴς υἱός in detail can be known in its full import only in the light of the whole of John’s proclamation. For ὁ μονογενὴς υἱός is simply a special form of ὁ υἱὸς τοῦ θεοῦ. When Jn. speaks of the Son of God, he has primarily in view the man Jesus Christ, though not exclusively the man, but also the risen and pre-existent Lord. The relation of the pre-existent Lord to God is that of Son to Father. This comes out indisputably in 17:5, 24. Jesus is aware that He was with God, and was loved by Him, and endued with glory, before the foundation of the world. This is personal fellowship with God, divine sonship. It is true that neither in the prologue, nor 8:58, nor c. 17 does Jn. use the term “son” for the pre-existent Lord. But He describes His relation to God as that of a son. To maintain that in Jn. the pre-existent Lord is only the Word, and that the Son is only the historical and risen Lord, is to draw too sharp a line between the pre-existence on the one side and the historical and post-historical life on the other. In Jn. the Lord is always the Son. Because He alone was God’s Son before the foundation of the world, because the whole love of the Father is for Him alone, because He alone is one with God, because the title God may be ascribed to Him alone, He is the only-begotten Son of God. [Büchsel, F. (1964–). μονογενής. In G. Kittel, G. W. Bromiley, & G. Friedrich (Eds.), Theological dictionary of the New Testament (electronic ed., Vol. 4, p. 741). Eerdmans.]
The only evidence Büchsel cited to support the Creed's claim that Jesus was "begotten" before creation is Jesus' prayer recorded in John 17 where He refers to God as "Father" and talks about "the glory which I had with You before the world was" (Jn 17:5) and "for You loved Me before the foundation of the world" (Jn 17:24). In my opinion, the fact that Jesus knew on earth that He was God prior to when He created everything is not enough evidence to disassociate "begotten" and the incarnation and move it to eternity past.