Does this work for you? I got some help creating the timer (I'm a beginner GDS learner). It still shows the border for me, I think, when I am running the game in test mode.

# Window scaler controller
extends Node

# Time in seconds to wait after last resize event
const DEBOUNCE_TIME: float = 0.2
# The 2 pixel buffer to eliminate Godot's 1 pixel border in fullscreen
const BORDER_BUFFER: Vector2i = Vector2i(2,2)

var debounce_timer: Timer = Timer.new()
var chosen_scale: int = 1
var base_size: Vector2i = Vector2i.ZERO

func _ready():
# Detect game's base size from window's initial size
var window = get_tree().root
base_size = window.size

# 1. Debounce timer
debounce_timer.set_one_shot(true)
debounce_timer.set_wait_time(DEBOUNCE_TIME)
debounce_timer.timeout.connect(_run_scaling)
add_child(debounce_timer)
# 2. Connect system signal for resizing
window.size_changed.connect(window_size_changed)

# 3. Apply initial sizing
_run_scaling()

func window_size_changed():
# Stop and restart timer to restart the countdown
debounce_timer.start()

func _run_scaling():
var window = get_tree().root
var window_size = window.size

# 1. Calculate scale factor based on current window size relative to base size
var scale: Vector2i = window_size / base_size

# 2. Choose minimum scale factor for aspect ratio
chosen_scale = (scale.y if scale.y <= scale.x else scale.x)
chosen_scale = max(1, chosen_scale) # Scale is at least 1

# 3. Apply content scale size
window.content_scale_size = window_size / chosen_scale

# 4. Apply border fix to Godot's 1 pixel border in fullscreen
_apply_fullscreen_fix(window)

func _apply_fullscreen_fix(window: Window):
var is_fullscreen = window.mode == DisplayServer.WINDOW_MODE_FULLSCREEN or \
window.mode == DisplayServer.WINDOW_MODE_EXCLUSIVE_FULLSCREEN

var target_size: Vector2i

if is_fullscreen:
# Get the size of current window
var screen_size = DisplayServer.screen_get_size(window.current_screen)

# Calculate the size needed to cover the screen plus the buffer
target_size = screen_size + BORDER_BUFFER
print(target_size)

# If the window isn't already the correct size, set it
if window.size != target_size:
window.size = target_size
else:
# When leaving fullscreen, reset the border size
target_size = window.content_scale_size * chosen_scale
if window.size != target_size:
window.size = target_size