Changed the code to set the position on _enter_tree and using process_priority to make sure all of the steps are done in the correct order, but this still doesn't produce the expected behavior. It's not clear to me when Godot renders the scene when you swap to it, because it seems nonsensical to say that it will render a scene before _enter_tree has been called for all of the nodes, because saying so would mean that it is rendering a scene that hasn't finished loading yet.

It could be possible that there is some kind of cached state, where even though the data has the correct values, the incorrect values have been sent to render. I'm not sure how to suss out this issue without more engine knowledge. crazy_stewie suggested I could use RenderDoc, which I think could be useful for diagnosing this specific potential issue.

I'd like to get a definitive answer on when exactly it will render a newly loaded scene since I feel like that will be a significant piece of understanding I should know. I'll have to do some investigation on this in the future.

crazy_stewie also suggested that I might also be able to use set_render_loop_enabled() to disable rendering until the new position is set. I think I might also be able to use a combination of set_render_loop_enabled(false) and force_draw() to do a step-by-step analysis of the problem.