Combine ketchup, mayonnaise, yellow mustard, and sweet relish in a bowl to make the burger sauce. Set aside.
Peel, wash, and cube the potatoes. Toss them with olive oil, salt, pepper, garlic powder, and paprika until well coated.
Cook bacon until crispy, preferably baked at 400°F for 20 to 25 minutes on a parchment-lined sheet pan, then crumble.
Cook the seasoned potatoes in the air fryer at 400°F for 20 to 25 minutes, shaking the basket occasionally to crisp evenly.
While the potatoes cook, heat a large oven-safe skillet over medium-high heat and brown the ground beef with chopped onion and seasonings.
Drain excess grease from the skillet, then stir in minced garlic and cook for about 30 seconds until fragrant.
Add the cooked potatoes to the skillet with the beef and mix everything together well.
Top the mixture with shredded cheese and crumbled bacon, then place the skillet in a 400°F oven until the cheese melts and bubbles.
Remove skillet from oven, drizzle burger sauce over the top, and sprinkle with chopped green onions. Serve with optional burger toppings.