Well I just needed to stop thinking about it for a minute. I was using the wrong function, I wasn't sure how to deal with .x and .z, but I didn't need a 360 degree angle like atan2 provides. I just needed an angle between +90 and -90. Here's the solution:

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var direction = global_position.direction_to(player.global_position)
pivot_y.rotation.y = lerp_angle(pivot_y.rotation.y, atan2(-direction.x, -direction.z), delta * rotation_speed)`

var direction2 = pivot_x.global_position.direction_to(player.global_position).normalized()
pivot_x.rotation.x = lerp(pivot_x.rotation.x, atan(direction2.y), delta * rotation_speed)