My impression from this is that the relationship between the staged user and the topic may be irrelevant for your purposes. One possibility that comes to mind is that you could allow moderators to change the owner of posts (moderators_change_post_ownership setting), if that’s appropriate for your forum, where part of the process would then be for the moderator to change ownership before moving the topic.
This might need to be tested though, I don’t know for sure how notifications for staged users are handled so potentially this won’t actually stop the staged user receiving emails for the topic.
Another option might be to write (or commission) a simple plugin to react to new topics in that private category, check if the author is staged and if so, do something. That might still be changing the owner but without requiring moderators have that permission and ensuring it won’t be forgotten or perhaps it’s possible to have the staged user unwatch the topic.
Edit: Upon re-reading, I realise that actually you will eventually want the email address from the staged user so changing the owner is probably unhelpful. I also notice there is the allow_changing_staged_user_tracking setting which suggests to me that staged users tracking/watching topics is how notifications are handled.