Hello, MlleSim, and welcome to the Grammar Exchange.
"No, something else in him had died, something that he had long desired should perish."
This is a structure which is correct + I know it is correct ⇒ This is a structure which I know is correct.
As you can see, we can merge a relative clause (which is correct) with a content clause (that it is correct) by saying "which I know is correct."
Your sentence is similarly formed:
It was something that should perish + He had long desired that it should perish ⇒ It was something that he had long desired should perish.