Rub the inside of a heavy pot or dutch oven with the cut side of the garlic. Add the hot sauce and bring to a simmer over medium heat.
In a bowl, toss the cheeses. Working in 3 batches, gradually stir the cheese mixture into the hot sauce until melted. Cook, stirring occasionally, until the mixture is gently bubbling, about 2 minutes; season.
Serve immediately from the dutch oven or transfer to a fondue pot. Serve with chicken tenders and celery sticks.