They are part of Israel. I thought that would be clear after reading God's conversation with Elijah. Paul does speak of those who are truly Israel, that is, the elect who will be saved by Christ. They are NOT part of the church, because they aren't saved. They are still unsaved. They are non-believers right now. That doesn't make them any less God's Ephesians 1 chosen people, the elect.
They will be saved at the end of the tribulation. Zechariah 12. Jesus will visit them personally. God will pour out repentance and grace on Israel, and they will look upon Him whom the have pierced, and they will mourn Him. They will ALL mourn Him, tribe by tribe, family by family. They are all the elect of God, and God is finally gathering them in. A national salvation of God's chosen elect of the nation of Israel.
You need to get that understanding.