Preheat oven to 400 degrees F (200 degrees C). Bring all the refrigerated ingredients to room temperature.
Slice the top off the garlic, exposing the cloves. Place in a square of aluminum foil and drizzle olive oil over garlic. Wrap foil around garlic.
Roast in the preheated oven until garlic is tender, 35 to 40 minutes. Cool roasted garlic completely.
Combine Camembert cheese, cream cheese, Parmesan cheese, Cheddar cheese, beer, and butter in a food processor; process until smooth. Add roasted garlic cloves, onion, caraway seeds, and paprika; process until smooth. The mixture may be a little on the "soupy" side but will firm up upon refrigeration; transfer to an air-tight container and refrigerate until flavors blend, at least 2 hours.