I changed it to set the position in one node and then read that position in the other and got it to break. It appears that _enter_tree and _ready are not affected by process_priority. In fact, although process_priority's documentation describes itself as changing the order in which nodes process the processing callbacks NOTIFICATION_PROCESS, NOTIFICATION_PHYSICS_PROCESS, and their internal counterparts, they really only mean _process and _physics_process. When the documentation refers to "internal counterparts", they are referring to NOTIFICATION_INTERNAL_PROCESS and NOTIFICATION_INTERNAL_PHYSICS_PROCESS. These NOTIFICATION_* constants are values passed to _notification(what), which are used by the low-level code in the engine.

I will never need to call _notification unless I need to use a notification that doesn't have a corresponding function or signal.

It looks like I can get the flickering to stop by using _enter_tree to set the game state and _ready to set the position.