You want to cover ME???? Why?!
If it were me, I wouldn't plan on building it then installing it. I'd plan on building it "in place" so to speak. Some kind of bevel would be my joint of choice, unless you can implement a lap joint into the look of it (like a reveal or overhang, or some kinda lip). I wouldn't rely on that joint to suspend the bottom piece alone, though. I would nail and maybe even glue the three boards onto the beam.
The reason I suggest doing it 'in place' instead is that you're surely going to have to scribe the two edges that meet with existing stuff. New or not, nothing's straight and you'll likely have to shape things to fit nicely. Also, this lets you account for any weird issues like a bow or sag or crook or some such crazy stuff in the beam itself.
As for the "boards are too short" issue, a "Scarf" joint is the way to go here. Find some really really closely matching stock and put the lowest angle you possibly can on their faces. I have trouble with this usually because my miter saw only goes to about 60 degrees and I'd like something tighter most of the time. But 60'd do in a pinch. The trick is to have good straight, square, and SMOOTH bevels on those ends that overlap. This way the seam is "spread" and made shallower.
Maybe stain and finish the boards first and then cut to fit on site?