Should I pre-cook sausages before scrambling into eggs?
It takes more heat/time to safely cook sausage than it does to fully cook eggs. This would be especially true for omelettes, where the egg is in contact with the hot pan, and the filling only warms indirectly. Add your expectation to start with cold sausage, and cooking the sausage and egg at the same time seems like a really bad idea. (Yes, you could just cook the whole assembly longer, but you'd end up with seriously vulcanized eggs.)

Be safe: cook the sausage, and then add the eggs.