Bring a pot of salted water to a boil over high heat. Add the linguine; cook until al dente, 8 to 10 minutes. Drain the pasta, reserving ¼ cup pasta water.

Meanwhile, heat the olive oil in a 12-inch skillet over medium-high heat. Add the green and red peppers, garlic, and onion; cook 3 minutes. Add the wine and the mussels; cook, covered, until the mussels open, about 2 minutes.

Add reserved pasta water, tomatoes, butter, and shrimp and cook, stirring, until the shrimp are just pink, about 1 minute. Add the cooked linguine, toss to combine, and cook, stirring occasionally, until the sauce thickens and clings to the pasta. Stir in the basil and season with salt and pepper.

Divide pasta between 2 bowls. Sprinkle with more basil and Asiago and garnish with lemon wedges.