Now, to be entirely fair, this is not, strictly speaking, impossible. It'd take a lot of explaining where the hell Cauldron got the superlative Occult knowledge of how to build the miniature Cage, plus explaining how the hell they got their hands on an Exaltation (what, do they just have Lytek's cabinet hanging in their garden shed, next to the Entity bits and the traumatized Case 53s?), but that bit isn't a total wash at least.
It's just. Once you release it, it's questionable if the Exaltation even has a definable position. It's certainly not under any obligation to attach itself to the one who opened the Cage. So there's no point.