Is Vessantara Jātaka in conflict with morality? Is there any convincing interpretation of Vessantara Jātaka that isn't in conflict with morality? Vessantara gave away his children to serve as slaves for no particular reason; it is not clear how not giving them away would hinder his own awakening.

I can see how one might argue from the Buddhist perspective that attachment to one's children and anger towards their oppressors is a bad thing. One can, however, protect one's children from oppressors in a completely detached way, without generating any attachment or hatred. Moreover, it's not that he just didn't resist; instead he actively looked for his children when they had run away, in order to give them away to Jūjaka.

All in all, Vessantara's behaviour caused a lot of suffering to his children for no reason whatsoever, and he not only didn't get condemned, but was even presented as the one who did the right thing. Is there any way to reconcile this story with morality?
At that time of the society husband is the sole owner of wife and children and he can give them away to anyone at will. To become a Buddha one needs to cultivate Dana Parami to the highest level. That includes the ability to give away anything you own. King Vessantara did not give away the children to be tortured. It was Jūjaka who decided to hit the children. So the Karmic consequences fall on Jūjaka. Not king Vessantara .

It was extremely tormenting for king Vessantara as you can see in the following passage:


  What follow are the verses of the Great Being's lamentation.
  
  "O when at morning or at eve for food my children cry, Opprest by
  hunger or by thirst, who will their want supply? [552] How will their
  little trembling feet along the roadway go, Unshod? who'll take them
  by the hand and lead them gently so?
  
  How could the brahmin feel no shame, while I was standing by, To
  strike my harmless innocents? a shameless man say I!
  
  No man with any sense of shame would treat another so, Were it a
  servant of my slave, and I brought very low.
  
  I cannot see him, but he scolds and beats my children dear, While like
  a fish caught in a trap I'm standing helpless here."
  
  These thoughts came into the Great Being's mind, through his affection
  for the children; he could not away with the pain to think how the
  brahmin cruelly beat his children, and he resolved to go in chase of
  the man, and kill him, and to bring the children back. But no, he
  thought: that was a mistake; to give a gift, then to repent because
  the children's trouble would be very great, that was not the way of
  the righteous. And the two following stanzas contain the reflexions
  which throw light on that matter.
  
  "He bound his sword upon his left, he armed him with his bow; I'll
  bring my children back again; to lose them is great woe. But even if
  my children die ’tis wicked to feel pain 1: Who knows the customs of
  the good, yet asks a gift again?........"


Also read the passage where the son defends his father's actions


  But the boy hearing this, could not stomach his father's blame; but as
  though raising with his arm Mount Sineru smitten by the windblast 1,
  he recited this stanza:
  
  "How, grandsire, can he give, when none in his possession are, Slaves
  male or female, elephants, a horse, a mule, a car?" The king said:
  
  [576] "Children, I praise your father's gift: no word of blame I say.
  But then how was it with his heart when he gave you away?" The lad
  replied:
  
  "All full of trouble was his heart, and it burned hot as well, His
  eyes were red like Rohinī, and down the teardrops fell."


So you can see that he gave them away not because he didn't care, but it was a bond that he had to overcome. What's ironic is that they say that the Mara could trouble the Buddha so much under the Bodhi tree because king Vessanatara lamented a lot for this great gift he gave. In comparison, when Maithree Bodhisatta becomes the Buddha, Mara can only watch from afar. He cannot come close enough to trouble the Buddha.