Does ισα το θεου refer to the role of Jesus as κυριos in Philippians 2:6?
In verse 10, Paul applied the monotheistic-YHWH text Isaiah 45:23 to Jesus which showed that Jesus was either co-equal with the κυριος of Isaiah 45 or the very κυριος himself of Isaiah 45. Either way, Jesus was being identified as divine in this text by giving him the role κυριος of the God of Israel. Thus, this passage acts similarly as 1 Corinthians 8:6 which is another text that equates Jesus with the God of Israel. This time it was specifically the κυριος of the Shema itself that was being ascribed to Christ [cf. Deut 6:4 LXX].
If Jesus was equal to God as κυριος in Philippians 2:6, then, that would mean that the kenosis of Jesus was about function, not essence. The rationale for this would be the next verse. Verse 7 says that Jesus emptied himself via taking the form of δουλος. The contrast between κυριος (master) and δουλος (servant) in Phil 2:6-7 would support that ισο το θεου in v. 6 to refer to Christ's role as κυριος. Is this plausible?
The word "kenosis" (kenosis) has entered theological language from Philippians 2:7, where in the sentence he "emptied himself" the Greek verb is ekenosen. "Kenosis," then, the corresponding noun, has become a technical term for the humiliation of the Son in the incarnation, but in recent years has acquired a still more technical sense, i.e. of the Son's emptying Himself of certain attributes, especially of omniscience (source: https://www.biblestudytools.com/dictionary/kenosis/).
In this question, the underlying theory is that the Son emptied himself of his ισα το θεου which is identified as his role as κυριος. In this case, it was function, not a certain divine attribute, that is being emptied by the Son. The lowly status of δουλος was the thing being substituted for the exalted status of κυριος. If this interpretation is correct, then, Phil 2:6 would be a parrallel to Phil 2:9-11