Summary
The translation offered by the OP best suits the entire passage:
and every tongue confess that [the] Lord Jesus Christ [is] in the glory of God the Father.
Paul begins the statement of glory in verse 6 by speaking of the appearance (glory) of God which Jesus emptied Himself of in order to take on the appearance (sin) of a slave. After His death, the God, ὁ θεὸς, exalted Him in the glory of God [the] Father, θεοῦ πατρός.
Paul has summarized and restated what the Gospel reports Jesus said on His final night:
9 Jesus said to him, “Have I been with you so long, and yet you have not known Me, Philip? He who has seen Me has seen the Father; so how can you say, ‘Show us the Father’? 10 Do you not believe that I am in the Father, and the Father in Me? The words that I speak to you I do not speak on My own authority; but the Father who dwells in Me does the works. 11 Believe Me that I am in the Father and the Father in Me, or else believe Me for the sake of the works themselves. (John 14 NKJV)
4 I have glorified You on the earth. I have finished the work which You have given Me to do. 5 And now, O Father, glorify Me together with Yourself, with the glory which I had with You before the world was. (John 17)
Jesus had the same glory as the Father which He emptied Himself of in order to take on the appearance of a slave. After the humiliation of death on the cross, His original glory was restored. Since Jesus has what He previously had, the glory of the Father, it is best to say He in the glory of God [the] Father.
morphē [not physis] theos - appearance of God
From Galatians we see morphē should be understood as something different from physis.
Formerly, when you did not know God, you were enslaved to those that by nature are not gods. (4:8 ESV)
Because Paul speaks of the nature in Galatians; he should be taken as saying something different in Philippians. morphē means form or external appearance so Paul is saying Jesus had the appearance of God (no article) and He did not consider it robbery to be equal to God (no article).
morphē doulos - appearance of a slave
Paul's meaning of morphē theos, appearance of God must be consistent with what he means when he repeats morphē in connection with doulos:
who, though he was in the morphē of God...taking the morphē of a slave
Was morphē theos became morphē doulos
Servant is doulos which can be translated as slave. So, according to Paul, Jesus had a morphē theos, which He gave up in order to take on a different morphē, which was morphē doulos.
Using Paul's letter to the Romans, the meaning of morphē doulos refers to the human condition:
16 Do you not know that if you present yourselves to anyone as obedient slaves, you are slaves of the one whom you obey, either of sin, which leads to death, or of obedience, which leads to righteousness? 17 But thanks be to God, that you who were once slaves of sin have become obedient from the heart to the standard of teaching to which you were committed, 18 and, having been set free from sin, have become slaves of righteousness. (Romans 6)
All humans are sinners (cf. Romans 3:23) and anyone who sins is a doulos to sin. Therefore, when Jesus was in a human body, His appearance was like any other human: He appeared to be a slave to sin. Of course, there was no sin in Him (cf. Hebrews 4:15).
In Romans, Paul personifies sin: the sinner is a slave to "Master" Sin. Jesus, on the other hand, has the appearance of a doulos. He is not a sinner and sin is not His master. Now, if one takes Paul as only narrowly focused on the sin nature of man, Philippians would mean:
He was the appearance of God, in other words He was without sin...and He took on the appearance of a slave, in other words, He looked like every sinner whose Master is Sin
morphē theos = the glory of the Father
A limited understanding of morphē theos as speaking only of God's sinlessness means Jesus, like God, was without sin. In this narrow sense translations like the NIV which treat morphē theos as the nature of God, are not wrong. However, that understanding does not make sense when seen in how Paul introduces the subject of morphē doulos:
but emptied himself, taking the form of a servant
Since He was already without sin, He could not empty Himself from sin. So then, what was the appearance He gave up in order to take on the appearance of a slave?
4 I have glorified You on the earth. I have finished the work which You have given Me to do. 5 And now, O Father, glorify Me together with Yourself, with the glory which I had with You before the world was. (John 17)
The appearance Paul is speaking of is glory which He had with the Father.
who, having the appearance of God [the glory of the Father] did not consider it robbery to be equal with God [the Father] so He emptied Himself [of this glory] and took on the appearance of a slave, coming in the likeness of men
Then Paul completes the passage by indicating what Jesus emptied Himself of was returned, to the glory of God the Father. Here Paul not only omits the article, He makes a further distinction between the glory of God and glory of the Father:
For all have sinned and fall short of the glory of God (Romans 3:23)
God here is written with the article, τοῦ Θεοῦ.
Conclusion
In Romans, Paul states all have sinned and fall short of the glory of God; in Philippians Paul states Jesus humiliation ends with glory of God Father. The two are not in contradiction because Jesus only took on the appearance of a slave (sinner). As such He did not fall short of the glory of God.
However, in order to take on the appearance of a sinner He had to empty Himself of His appearance as God. That is, He gave up the glory He had with the Father from the before the world was. In His appearance as a slave, He did not fall short of the glory of God (with the article); He was simply without the glory of God Father since He had emptied Himself of that appearance.
Paul's letter to the Philippians is an explicit statement the request Jesus made, glorify Me together with Yourself, with the glory which I had with You before the world was was answered.
