My humble opinion..
The beam would be more effective and less trouble than sistering 2x12s in. When you start trying to sister in the joists, you will find that you will have to remove a portion of the roof and slide them in from outside. It is physically impossible to get a ceiling joist in the space from inside. You will get one end in a space, then the other end will hit the rafters and joists and will not make the turn to drop into place. Trust me on that
Even at 12' span, the beam would be effective. The 2x6's though would be marginal for a 12' span in a living area.
But, houses built before 1970 often have true 2x6 floor/ceiling joists. It's been done before..
Make sure you provide adequate support under the ends of the beam. You will need at least 3 studs under each end and make sure the studs are sitting on a framing member under the floor. Bridging from the exterior wall to the next joist, or center it over a floor joist.
Anyway.. good luck!
Welcome to the forum!