I would do my best to get a beam in the center of the span; cutting it down to a 10' span on each side, I could live with that. Then, I would also install my side walls in a way that would tie the joists to the rafters which would shorten the span even more and stiffen things up considerably. Then, I would try to use the lightest materials available in the attic. For instance, 3/4" tounge and groove underlayment grade plywood glued with PL 400 and screwed every foot, mimimum (extra layers or the osb type underlayment would be much heavier). I would insulate the roof space and the walls of the new attic and cover both with 1/4" luan paneling, glued and nailed. For a finish on the floor, maybe some laminate flooring.
I would still check with the steel supplier on the beam using the 3/8" steel plate sandwiched between 2 2 Xs with lots of bolts and support it on each end with the 3 2 X 4s which would be bearing on outside walls or piers to support the floor under the end posts. I would make the access to the attic by a spiral staircase, there is a steel one that would add very litttle weight made by The Iron Shop and advertised in The Mother Earth News.
I am still open to questions or other suggestions.