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Well, see, there's a problem already. God was not the God of Ishmael.

False; both the Hebrew and Genesis indicate otherwise.

Both Hebrew and Arabic combine their language's respective word, not name, for God to form names. In the case of a Hebrew, El means God and in Arabic, alLa means "The God." Ishmael's name means "God will hear." The angel of the LORD chose his name.

And the angel of the Lord found her by a fountain of water in the wilderness, by the fountain in the way to Shur. And he said, Hagar, Sarai's maid, whence camest thou? and whither wilt thou go? And she said, I flee from the face of my mistress Sarai. And the angel of the Lord said unto her, Return to thy mistress, and submit thyself under her hands. And the angel of the Lord said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude. And the angel of the Lord said unto her, Behold, thou art with child and shalt bear a son, and shalt call his name Ishmael; because the Lord hath heard thy affliction. And he will be a wild man; his hand will be against every man, and every man's hand against him; and he shall dwell in the presence of all his brethren. And she called the name of the Lord that spake unto her, Thou God seest me: for she said, Have I also here looked after him that seeth me? Wherefore the well was called Beerlahairoi; behold, it is between Kadesh and Bered. And Hagar bare Abram a son: and Abram called his son's name, which Hagar bare, Ishmael. And Abram was fourscore and six years old, when Hagar bare Ishmael to Abram.
Genesis, Catholic chapter sixteen, Protestant verses seven to sixteen,
as authorized, but not authored, by King James