In a small bowl, combine mayonnaise, horseradish, and ketchup and set aside.
Preheat grill to high heat. Rub onion and pineapple slices with olive oil and grill 3 to 4 minutes per side or until tender, but still holding their shape, then set aside. Meanwhile, season burger patties with salt and pepper and grill for 8 minutes per side for medium rare, or about 12 minutes per side for medium, or longer depending on desired doneness.
Let rest 4 minutes before assembling burgers.
Divide horseradish mayonnaise between 4 bottom buns. Top with lettuce, burger, grilled onion and pineapple. Add top bun and serve immediately.