Shouldn't Ur-Quan Kohr-Ah also have Battle Thralls?

This conundrum immediately reminded me of the way the tributes in the Hunger Games teamed up into groups to work together even though there could only be one winner. They'd ultimately have to turn on one another for there to be a champion. I suppose you could say they did it to survive just a little longer and perhaps avoid a brutal death at the hands of a more vicious enemy, but the story never seemed to pay much attention to addressing that point.

I think your question is at the heart of their Doctrinal Conflict with the Kzer-Za. The Kohr-Ah were the warrior caste, so yes, they essentially viewed everyone else as inferior or insignificant, perhaps even their Kzer-Za brethren to a degree. The Kzer-Za wanted thralls and the Kohr-Ah did not, which was one of the core motivations of their civil war.

The idea of a marauding alien species is nothing new in science fiction, but I always thought the warped religious zealotry of the Kohr-Ah was a nice element that made them kind of unique and really intimidating simply because they seemed so cool and calm about their obvious psychosis, as though they'd really thought things through and decided that this was their best course of action. Who's to say you wouldn't feel the same way after a few thousand years of psychic enslavement?

I think the bits of conversation where the Kohr-Ah offer the genocide-suicide option was in the spirit of humor as most of the writing was, although I suppose if another race had accepted that offer they would by definition be thralls, but I'm not sure why it would be strange that no other race accepted; why help the Kohr-Ah in their genocide if they're ultimately going to destroy you for your efforts anyway? Might as well go down fighting them rather than someone else.

Indeed individuals, in such alliances, in Hunger Games and Battle Royale somehow did not create contingency plans for when only one alliance remains. Perhaps they still hoped the rules could still change.

"why help the Kohr-Ah in their genocide if they're ultimately going to destroy you for your efforts anyway? Might as well go down fighting them rather than someone else."
Answer: In both World Wars, superior officers would shoot troops refusing to attack their enemy. If what you said was true, then troops that faced impossible odds would rather have shot their own officers. Which on occasion may have happened, but I doubt the propaganda campaign would let out such information.

But yes the Kohr-Ah having Battle Thralls would contradict that they have no use for what they deem inferior races:

Captain: What if we promise to be your slaves? Will you let us live then?

Kohr-Ah: The ignominy of slaving we leave to our Kzer-Za cousins.
We have no need for inferiors as servants.

Kohr-Ah are also content with death and believe in reincarnation, so casualty count on their own side doesn't worry them.

I would get the genocide-suicide black humor, if they offered the other race to commit suicide as the only alternative to Kohr-Ah killing them instead. Humane irony.