The description I read, says that if left alone, they merge back into one. To me this suggests the two halves are part of the original creature.
If you split the Jelly in two and halve the remaining hits points, presumably you have to halve the max hit points as well? Therefore the two halves are still bloodied. Just not bloodied for the first time.
Personally I would allow any power that triggers on zero hit points to trigger on zero hit points and any power that triggers on the death of the original Jelly to trigger when the last half is destroyed. But then , as a GM, I am a total softie