Additionally, we are told in Scripture that the TRUE Father of Jesus is God (Isaiah 7:13 - 14, Psalm 2:7, etc.). Would not this fact also mean that he could NOT be the Savior since he was not a Jew?
ANSWER: The basic issue in your questions is whether Jesus' status as a tribe member is based on having a father who is a Jew. Doubts about his true parentage and lineage have been an issue since the first century A.D.
When Jesus walked the earth 2,000 years ago, most people did not accept the claim that a virgin could have miraculously given birth to him. This disbelief reached a crescendo during his ministry when some religious leaders accused him of being born illegitimately (e.g. Mary conceived him through sex outside of marriage).
Then they (self-righteous Pharisees who wanted him dead) said to Him (Jesus), "We have not been born of fornication (like we believe you have!). We have one Father, and that is God" (John 8:41, HBFV throughout unless stated)
Even today, some Bible translations like the Revised Standard Version, Jewish Publication Society Bible, and others have REMOVED from Isaiah 7:14 any reference to the Lord being born to a virgin!
Therefore the Lord himself will give you a sign. Behold, A YOUNG WOMAN shall conceive and bear a son . . . (Isaiah 7:14, RSV)
Getting back to your original question, for those who descended from the tribe of Judah, the descent of the father (paternal) is not as important as that of the mother (maternal). The mother is the decisive factor in determining whether a child is considered a Jew or not.
For example, a baby born with a mother from the tribe of Judah but a Gentile (non-Israelite) father is automatically a Jew. However, a baby produced by a father who is from the tribe of Judah and a mother who is a Gentile (non-Israelite) is not.
A tale of two lineages
Hebrews, in general, did not include women in genealogical tables. How, then, was Mary's lineage included in the Bible? The gospel writer Luke resorted to a little 'trick' in order to record that the bloodline of Jesus, through Mary, went all the way back to King David, Judah (one of the sons of the patriarch Jacob) and then all the way back to Adam. Luke begins tracing our Savior's physical line by stating Mary's husband, Joseph, was the son of Heli (Luke 3:23). Joseph's father, however, according to the book of Matthew (Matthew 1:16), is not named Heli but JACOB. Heli is the name of MARY'S father (the father-in-law of Joseph). Luke uses Joseph's name to represent, symbolically, Mary in the lineage he records. Jesus, therefore, is a Jew by blood through his mother.
The book of Matthew, chapter 1, traces the descendants of Abraham to Judah, then King David, then to Joseph. What does Matthew's lineage show since, as you rightly state, Jesus' true Father was God in heaven (Matthew 1:18, 20, Luke 1:34 - 35)? It shows our Savior's LEGAL lineage, which means he has a legal right to be considered a son of David. This claim is asserted in the very first verse of the New Testament.
1. The book of the genealogy of Jesus Christ, THE SON OF DAVID, the son of Abraham (Matthew 1:1)
Our Savior's pedigree is also asserted in other places in the New Testament such as the following.
2. Which He had promised before through His prophets in the Holy Scriptures 3. Concerning His Son; Who came from the seed of David according to the flesh (Romans 1:2 - 3)
14. For it is quite evident that our Lord has descended from Judah, of which tribe Moses spoke nothing concerning the priesthood (Hebrews 7:14, see also Acts 13:22 - 23)
Jesus is both, by blood and legally, a Jew and a descendant of King David. It was Joseph's lineage back to David that forced him and a very pregnant Mary to travel to Bethlehem. It is in Bethlehem that our Savior is born (Luke 2:4, 6 - 7).