Is the female questioner's shaming of her relative and her daughter considered a tit-for-tat response, or is it a sin that necessitates regret and expiation, especially since the relative and her daughter have not forgiven her?

Taunting a Muslim with an illness or affliction they have suffered is a forbidden form of harm. If a woman taunts you about your illness, it is not permissible for you to taunt her daughter about what has befallen her.

You must repent and seek forgiveness, and that woman must also repent for taunting you.

It was permissible for you to defend yourself by saying that she is sinful and disobedient to Allah by her taunting.

You must apologize to her daughter. If she forgives you, then praise be to Allah. If not, then you have done what was required of you, and your repentance along with your apology is sufficient for you.

And explain to her that she is obliged to accept your apology, because Allah does not burden a soul beyond its capacity.