My suspicion for why Julian didn't put the necklace back on is that he was trying to turn her into Killer Frost. When he went to kiss her earlier and she flinched and questioned his bedside manner, maybe that was him reopening up the wound or aggravating it so that it would cause precisely this situation where she begins to crash and die. And then when she does, he knows that she can't be shocked back to life so it looks like there's no choice but to pull off her necklace, as if he's doing it in the heat of the moment.
But maybe we'll see when the show returns another point of view where we see what he was really doing. Maybe Julian still is Alchemy and working for Savitar (either consciously or unconsciously) and is the one to betray the team.
I've suggested before that Savitar could actually be Julian all along, because otherwise there's no-one else in this season whom Savitar could be revealed as. Maybe the whole Alchemy schtick was all a pretence, and he is both Alchemy and Savitar. And the reason he's several steps ahead is not because he knows the future as such, but because he's right there with the rest of the team and listening to their plans on how they intend to stop Savitar.
So maybe he's actually trying to make the prophecies come true. He could have pushed Caitlin to actually become Killer Frost rather than allowing her to turn into her herself. And so maybe he withheld the necklace from her rather than putting it back because he knew precisely what he was doing.