Melt butter in a skillet over medium heat. Swirl olive oil into melted butter. Cook garlic, shallots, and capers in oil and butter until shallots are translucent, about 5 minutes. Mix in tomato sauce, Italian seasoning, and red pepper flakes; stir until well incorporated. Cover and reduce heat to medium-low; allow to simmer for 10 minutes.
Add mussels to the pan and cover tightly. Raise heat to medium-high and cook until all mussels have opened, about 10 minutes. Remove from heat, garnish with chopped green onions to serve. Discard any unopened mussels.