I felt the answer was obvious; my husband told me to write it up so here goes...
Reuben offers his two sons (Jacob's grandchildren) as sureties. It seems to be his guilt speaking (for he intended to save Joseph but instead is party to causing him to be sold, possibly lost or dead) when he declares:
|לז וַיֹּאמֶר רְאוּבֵן, אֶל-אָבִיו לֵאמֹר, אֶת-שְׁנֵי בָנַי תָּמִית, אִם-לֹא אֲבִיאֶנּוּ אֵלֶיךָ; תְּנָה אֹתוֹ עַל-יָדִי, וַאֲנִי אֲשִׁיבֶנּוּ אֵלֶיךָ.||37 And Reuben spoke unto his father, saying: 'Thou shalt slay my two sons, if I bring him not to thee; deliver him into my hand, and I will bring him back to thee.'|
Jacob is disturbed by this offer- these are the grandiose, desperate words of an individual who has not actually suffered child-loss but who is attempting to atone. If one understands the loss of a child (as Jacob has experienced, losing Joseph) they would never make a statement like this. It would be unthinkable.
Judah, in contrast, has lost children. Two of them. And he understands what it is like to withhold a third child out of fear of losing him as well. Indeed, there's a distinct parallel between him and his father, Jacob. In Genesis 38:11 Judah says
|יא וַיֹּאמֶר יְהוּדָה לְתָמָר כַּלָּתוֹ שְׁבִי אַלְמָנָה בֵית-אָבִיךְ, עַד-יִגְדַּל שֵׁלָה בְנִי--כִּי אָמַר, פֶּן-יָמוּת גַּם-הוּא כְּאֶחָיו; וַתֵּלֶךְ תָּמָר, וַתֵּשֶׁב בֵּית אָבִיהָ.||11 Then said Judah to Tamar his daughter-in-law: 'Remain a widow in thy father's house, till Shelah my son be grown up'; for he said: 'Lest he also die, like his brethren.' And Tamar went and dwelt in her father's house.|
This echoes Jacob's fear, declared in Genesis 42:36
|לו וַיֹּאמֶר אֲלֵהֶם יַעֲקֹב אֲבִיהֶם, אֹתִי שִׁכַּלְתֶּם: יוֹסֵף אֵינֶנּוּ, וְשִׁמְעוֹן אֵינֶנּוּ, וְאֶת-בִּנְיָמִן תִּקָּחוּ, עָלַי הָיוּ כֻלָּנָה.||36 And Jacob their father said unto them: 'Me have ye bereaved of my children: Joseph is not, and Simeon is not, and ye will take Benjamin away; upon me are all these things come.'|
Judah lost two children and sought to protect the third.
Jacob loses two children (Joseph, Simeon) and seeks to protect the third.
And thus, when Judah speaks to Jacob, he comes from a place of understanding and empathy. He understands Jacob because he has lived what has happened to Jacob. Importantly, he has also learned how to take responsibility for his actions- good and ill. As Rabbi Ari Kahn explains, Judah is the one to coldly say "What profit is there in our brother's death?" and to suggest the sale instead. It is therefore unsurprising that his own children show no familial leanings- they do not wish to build a family lest they mar their wife Tamar's beauty and they do not wish to build up their dead brother's legacy through siring a child in his stead. When Judah declares to Tamar, "You are more righteous than me," is is the beginning of an understanding that overall, he has lived his life incorrectly. First, he witnesses the lengths to which she is willing to go to build a family, and feels shamed by his own indifference to his family. Second, the man who coldly weighed the profit in a brother's murder is now willing to step forward to save Tamar from the devouring fire even though she wronged him, deceiving him and now the cause of his public humiliation (the rabbinic equivalent of death). This choice- to put family and life above his personal gratification- is the turning point that will eventually lead him to step forward in Parshat Vayigash and offer himself in place of Benjamin.
And thus it makes sense that Judah declares (Genesis 43:8-9)
Let us live and not die- we, you and our little ones- Judah's focus is on the value of family- all family, no matter whether they profit him or do not profit him.
I will be a surety to him- I have already demonstrated that I will put the life of another above my own, as shown by the episode with Tamar.
Of my hand shall you require him- I, who understands what it is like to lose two children, and who could not bear to part with my third.
If I bring him not to thee and set him not before thee, then let me bear the blame forever- I will know how I have failed you in a way that no other can know. Because I know what it is like for a child- two children- to die, and there is no blame you could apportion to me that would be worse than the blame I would feel if I caused you to lose this child.
Jacob can trust this vow, and so he allows Benjamin to travel with Judah.