In a large bowl, toss the cheeses with the cornstarch. Rub the inside of a 3-quart fondue pot with the garlic; discard.
In the fondue pot, heat 1 cup wine with the lemon juice over medium-low heat until steaming but not yet simmering. Stir in the cheese mixture, 1 handful at a time, and cook until melted. If the fondue is too thick, gradually stir in up to 1/2 cup more wine. Season to taste with white pepper. Serve with the bread cubes.