If Sarah's anger was only toward Ishmael, then could Sarah have halakhically asked Abraham to only expel Ishmael?

Question

If Sarah's anger was only toward Ishmael, then could Sarah have halakhically asked (or within her rights, from whichever source, to ask) Abraham to only expel Ishmael and not Hagar, seeing Hagar is owned by Sarah?

(Yes, I understand this entire event took place before Halakha or Torah came into existence. I will therefore accept answers, as if though Halakha were applicable.)

וַתֹּ֨אמֶר֙ לְאַבְרָהָ֔ם גָּרֵ֛שׁ הָֽאָמָ֥ה הַזֹּ֖את וְאֶת־בְּנָ֑הּ כִּ֣י לֹ֤א יִירַשׁ֙ בֶּן־הָֽאָמָ֣ה הַזֹּ֔את עִם־בְּנִ֖י עִם־יִצְחָֽק:

I am also trying to find out if Sarah was also being kind to Ishmael by allowing Ishmael's mother to go with him, as Ishmael would be totally helpless at that age; at the same time, was it also, in part, an act of kindness to Hagar that Sarah is setting her free, if the understanding that Hagar is a slave of Sarah is correct?

Answer 1

Sforno says that Yishmael's taunting was initiated by his mother, Hagar.

גרש האמה הזאת ואת בנה, seeing that what the son did was at the instigation of his mother. He spread lies about Yitzchok’s legitimacy in order to establish a claim to your inheritance.

(Translation from Sefaria.org)

Rashi also comments on the previous verse and says that Yishmael had fallen into idolatry.

מצחק MAKING SPORT — This means worshipping idols, as it is said in reference of the Golden Calf, (Exodus 32:6) “And they rose up to make merry (לצחק).” Another explanation is that it refers to immoral conduct, just as you say in reference to Potiphar’s wife, (Genesis 39:17) “To mock (לצחק) at me.” Another explanation is that it refers to murder, as (2 Samuel 2:14) “Let the young men, I pray thee, arise and make sport (וישחקו) before us” (where they fought with and killed one another) From Sarah’s reply — “for the son of this bondwoman shall not be heir with my son״ — you may infer that he (Ishmael) was quarrelling with Isaac about the inheritance, saying, “I am the first-born and will, therefore, take a double portion”. They went into the field and he (Ishmael) took his bow and shot arrows at him (Isaac), just as you say (Proverbs 26:18-19) “As a madman who casteth firebrands, [arrows and death] and says: I am only מצחק mocking” (Genesis Rabbah 53:11).

(Translation from Sefaria.org)

With Hagar being from Egypt, it would be a likely assumption that the only way Yishmael could have been introduced to idolatry in Avraham's camp would be through her.

Additionally, I have a hard time imagining Hagar being okay with staying behind with Avraham when Yishmael would be sent away.

Answer 2

Rabbi Hertz among others point out that the sending away of Yishmael and his mother was the official method at that time of disinheriting the child of a concubine. This is similar to the gifts given to the children of Keturah before Avraham died. Note, that this occurred when Yishmael was about 15 or so, while he did maintain ties with Avraham and his family.

Thus, this was a matter of ensuring that Yitzchak would be the only legal heir of Avraham. He could not have just banished Yishmael. As Sarah said

And Sarah said to Abraham, "Drive out this handmaid and her son, for the son of this handmaid shall not inherit with my son, with Isaac."

And Hashem told Avraham

And God said to Abraham, "Be not displeased concerning the lad and concerning your handmaid; whatever Sarah tells you, hearken to her voice, for in Isaac will be called your seed.

This was a legal maneuver only as we see later:

Rashi Vayeira 22:3 shows that 35 years later Yishmael was with him when he took Yizchak to the Akeidah

his two young men: Ishmael and Eliezer, for a person of esteem is not permitted to go out on the road without two men, so that if one must ease himself and move to a distance, the second one will remain with him. — [from Pirkei d’Rabbi Eliezer, ch. 31; Gen. Rabbah ad loc., Tan. Balak 8]

Chayei Sarah 25:9 shows that Yishmael was at the funeral and acknowledged Yitzchak as the heir.

9 And Isaac and Ishmael his sons buried him in the Cave of Machpelah in the field of Ephron the son of Zohar the Hittite, which faces Mamre,

Rashi

Isaac and Ishmael: (Gen. Rabbah 30:4, 38:12) From here [we may deduce] that Ishmael repented and let Isaac go before him, and that is the meaning of “a good old age” which is stated regarding Abraham (above 15:15). - [B.B. 16b]