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This is a good question -- or rather, set of questions. I begin by reiterating a comment from the Q&A linked by OP: to engage with this set of issues fully, one really needs to consult Catrin H. Williams, I Am He: The Interpretation of ʾAnî Hûʾ in Jewish and Early Christian Literature (WUNT II/113; Mohr Siebeck, 2000). There is plenty of other relevant scholarship, but engaging Williams's work first-hand is mandatory to grapple with this question. A StackExchange Q&A will only take us so far!

As noted by OP, in that previous Q&A I cited C.K. Barrett's opinion that "...ἐγώ εἰμι is in itself (as Greek) a meaningless expression". Wider reading suggests there are other well-informed views to take into account which help considerably here.

This is a good question -- or rather, set of questions. I begin by reiterating a comment from the Q&A linked by OP: to engage with this set of issues fully, one really needs to consult Catrin H. Williams, I Am He: The Interpretation of ʾAnî Hûʾ in Jewish and Early Christian Literature (WUNT II/113; Mohr Siebeck, 2000). There is plenty of other relevant scholarship, but engaging Williams's work first-hand is mandatory to grapple with this question. A StackExchange Q&A will only take us so far!

As noted by OP, in that previous Q&A I cited C.K. Barrett's opinion that "...ἐγώ εἰμι is in itself (as Greek) a meaningless expression". Wider reading suggests there are other well-informed views to take into account which help considerably here.

Is ἐγώ εἰμι testified in extra-biblical Greek...?

Given the context, I suppose this should be slightly filled out as:

Is the absolute use of ἐγώ εἰμι testified in extra-biblical Greek...?

Is the absolute use of ἐγώ εἰμι testified in extra-biblical Greek...?

1. The Hebrew construction itself suggests a formulation in which the predicate is latent. This is the implication of Joüon-Muraoka, Grammar of Biblical Hebrew, § 154j (see p. 540, at end of that section; § 154i also informs this view).
2. In the appendix devoted to ἐγώ εἰμι at the end of vol. 1 of his Anchor Bible John commentary,⁴ Raymond Brown expresses the view that it is unexceptional, "a phrase of common speech" (p. 533, a "spectrum of usage, extending from the banal to the sacral..."). On p. 536 he has a discussion of the Isaiah texts which intersects with the Joüon-Muraoka material, noting also the example of Isaiah 45:18, where MT's אֲנִי יהוה is found as ἐγώ εἰμι in the LXX.

This is a small sampling of some of the key earlier studies with which Williams interacts on this question, with such online sources as I have been able to find. The earliest identification of ʾănî hûʾ as the "source" for the NT's absolute uses of ἐγώ εἰμι is credited to F.A. Lampe, Commentarius in Evangelium Joannis (1726). Fundamental developments were contributed in the early 20th C by Ethelbert Stauffer, seen in an early form in the ἐγώ εἰμι section of his larger ἐγώ article in G. Kittel (ed.), Theological Dictionary of the New Testament (Eerdmans, 1964; German 1935), pp. 352-354, but further developed in later publications. A yet more full and nuanced set of studies came from Stauffer's student, Johannes Richter, Ani hu und Ego eimi: die Offenbarungsformel 'Ich bin es' im Alten und Neuen Testament (1956). Contemporary, but working independently, Heinrich Zimmermann produced a dissertation in 1953 in Bonn, "Das absolute 'Ich bin' als biblische Offenbarungsformel" with two summary articles following in 1960. According to Williams, Zimmermann develops the LXX material to a greater extent than his predecessors. In English, a key study was contributed by Philip Harner, The "I am" of the Fourth Gospel: A Study in Johannine Usage and Thought (Fortress, 1970).

1. Those interested should note that a subset of the TLG is available for consultation for those without personal or institutional access to the full database. The survey reported briefly above used the full TLG.
2. Williams, I Am He, pp. 11-12 and also see footnotes 50, 51 on those pages for full details and any qualifications and "near misses".
3. Williams quotes Marguerite Harl as claiming that "the MT employs the 'virtually untranslatable' אני הוא (Deut. 32:39a: 'moi, moi, lui') as a divine name...". Williams, p. 58; M. Harl, "Le grand cantique do Moïse en Deuteronome 32...", in La langue de Japhet: Quinze études sur la Septante et le grec de chrétiens (Éditions du Cerf, 1992), p. 131.
4. R.E. Brown, The Gospel according to John I-XII (Yale Anchor Bible 29; 1995; first edn, 1966), pp. 533-538.

1. The Hebrew construction itself suggests a formulation in which the predicate is latent. This is the implication of Joüon-Muraoka, Grammar of Biblical Hebrew, § 154j (see p. 540, at end of that section; § 154i also informs this view). It is plausible to see the Greek formulation also carrying an "understood" predicate (as suggested by OP), although this does not work in every case. It is nonetheless interesting that the Hexaplaric witnesses do not ever appear to stumble over this construction, or feel compelled to fill it out.⁴
2. In the appendix devoted to ἐγώ εἰμι at the end of vol. 1 of his Anchor Bible John commentary,⁵ Raymond Brown expresses the view that it is unexceptional, "a phrase of common speech" (p. 533, a "spectrum of usage, extending from the banal to the sacral..."). On p. 536 he has a discussion of the Isaiah texts which intersects with the Joüon-Muraoka material, noting also the example of Isaiah 45:18, where MT's אֲנִי יהוה is found as ἐγώ εἰμι in the LXX.

This is a small sampling of some of the key earlier studies with which Williams interacts on this question, with such online sources as I have been able to find. The earliest identification of ʾănî hûʾ as the "source" for the NT's absolute uses of ἐγώ εἰμι is credited to F.A. Lampe, Commentarius in Evangelium Joannis (1726). Fundamental developments were contributed in the early 20th C by Ethelbert Stauffer, seen in an early form in the ἐγώ εἰμι section of his larger ἐγώ article in G. Kittel (ed.), Theological Dictionary of the New Testament (Eerdmans, 1964; German 1935), vol. 2, pp. 352-354, but further developed in later publications. A yet more full and nuanced set of studies came from Stauffer's student, Johannes Richter, Ani hu und Ego eimi: die Offenbarungsformel 'Ich bin es' im Alten und Neuen Testament (1956). Contemporary, but working independently, Heinrich Zimmermann produced a dissertation in 1953 in Bonn, "Das absolute 'Ich bin' als biblische Offenbarungsformel" with two summary articles following in 1960. According to Williams, Zimmermann develops the LXX material to a greater extent than his predecessors. In English, a key study was contributed by Philip Harner, The "I am" of the Fourth Gospel: A Study in Johannine Usage and Thought (Fortress, 1970).

1. Those interested should note that a subset of the TLG is available for consultation for those without personal or institutional access to the full database. The survey reported briefly above used the full TLG.
2. Williams, I Am He, pp. 11-12 and also see footnotes 50, 51 on those pages for full details and any qualifications and "near misses".
3. Williams quotes Marguerite Harl as claiming that "the MT employs the 'virtually untranslatable' אני הוא (Deut. 32:39a: 'moi, moi, lui') as a divine name...". Williams, p. 58; M. Harl, "Le grand cantique do Moïse en Deuteronome 32...", in La langue de Japhet: Quinze études sur la Septante et le grec de chrétiens (Éditions du Cerf, 1992), p. 131.
4. See F. Field, Origenis Hexaplorum, 2 vols (1875), on Deut 32:39 or the Isaiah references.
5. R.E. Brown, The Gospel according to John I-XII (Yale Anchor Bible 29; 1995; first edn, 1966), pp. 533-538.