That Klingon culture isn't a monolith over their entire history.
I don't see any inconsistency between the death yell and that the body is trash. The yell is to warm of the Warrior's spirit's assent to Grethor. The body no longer has the spirit and is just an empty meat shell.
Other Klingons may acknowledge that the body is empty, but still don't want the spirit of the dead to be dishonored by having its body eaten by slowly scavengers, something famously uncommon on starships. Just like how some humans believe in Sky Burials and others believe the body must go in a tomb the same day it died.
Or, my pet theory, Worf Lied to make Miles feel better about Enrique's death. It wouldn't be the first time he made up a "super serious" Klingon ritual to prove his point.
I think this would be a great topic to place in c/daystrominstitute if it's still active. It's been years since I've seen this debate online.