It seems doubtful Rex nor Shōchū will be going into the RoSaT at this point. Rex because it's doubtful he'll be welcomed enough and Shōchū because she'll probably want those native of the timeline to be able to focus on training instead of her taking up such valuable resources.

As for what either will be doing? I dunno; the main plot doesn't really have any hooks for either of them just yet.