Start by draining the canned tuna and breaking it up with a fork.
Add the mayo, lemon juice, sweet relish, mustard, salt, and pepper, and mix well. Check for seasoning. If the tuna is a bit dry, add more mayo.
Spread about 1 tbsp of softened butter on one side of each slice of bread.
Add two slices of cheese on each slice of bread, add about 1/2 cup of the tuna salad on top of the bread, and then cover with the other slice of bread.
Place the tuna melt sandwich in the air fryer basket butter-side-down and cook at 400 degrees F for 5 minutes. Once the 5 minutes are up, flip the sandwich and cook for an additional 3 minutes.
Notes
The bread slices I used for this recipe were pretty big so it took 9 minutes total to cook perfectly. If using white sandwich bread, check the tuna melt at the 4-minute mark because it may cook a lot quicker.