Whisk the eggs, 1/2 teaspoon salt, and 1/8 teaspoon pepper together in a large bowl. Set aside.
Fry the bacon in a 12-inch nonstick skillet over medium-high heat until crisp, about 8 minutes. Stir in the tomato and bell pepper and cook until the vegetables are softened, about 6 minutes.
Stir in the garlic and cook until fragrant, about 30 seconds. Transfer the mixture to a paper towel-lined plate and wipe the skillet clean with paper towels. Return the skillet to medium heat.
Melt the butter in the skillet, swirling to coat the skillet, until the foaming subsides. Add the eggs and bacon mixture to the pan and cook, stirring gently in a circular motion, until the mixture is slightly thickened, about 1 minute.
Use a rubber spatula to pull the cooked edges of the egg toward the center of the pan, tilting the pan so the uncooked egg runs to the cleared edge of the pan. Repeat until the bottom of the omelet is just set but the top is still runny, about 1 minute.
Cover the skillet, reduce the heat to low, and cook until the top of the omelet begins to set but is still moist, about 5 minutes.
Remove the pan from the heat. Sprinkle the cheese evenly over the omelet, cover, and let it sit until the cheese partially melts, about 1 minute. Slide half of the omelet onto a serving platter using a rubber spatula, then tilt the skillet so the remaining omelet flips over onto itself, forming a half-moon shape. Cut the omelet into wedges and serve immediately.