It's flat plate, obtained from your local ironmonger...make each leg 3 ft was what was intended altho I see as I wrote it, it does sound like meant overall.
Or, of course, you can sister another tubaX the same way or use exterior ply for the splice with a cap to hide the exposed edge.
The metal solution would, with only a little extra effort/expense, let you have the pieces fabricated with an angle so the center could be higher than the ends that could lend a nice aesthetic touch. Need a buddy who can weld or have the metal shop do it for you.
I don't recall now precisely what it ran, but we put a 12-ft 1/2" x 8" plate across the 10-ft span of the barn driveway after we jacked the sag out of the middle of the span when reroofed and repainted/repaired the old barn here a few years ago. That piece was under $200 as I recall new; if the scrap dealer in town had happened to have a piece at the moment I called it could've been much less. He's (the scrap man, that is) is really complaining these days that steel prices are way off, so while I've not priced anything recently, it shouldn't be too bad. Again, go to an iron/metals monger, not the box stores.
And, two 3/8" are probably overkill; two 1/4 or 5/16" would likely suffice as they're constrained from flexing sideways by being bolted to the beams it's the vertical height that is key dimension; same effect as the 2x8 vis a vis the 4x6 for stiffness in the vertical direction; the section is still rectangular x-section so same formula for I.
Actually, of course, you could make the same arch construct with the ply gussets as well. I've never tried it; don't know how well the pressure treated would bond with resorcinol to do a glue laminated joint; you might try to do some research and see if anybody has any suggestions on that. The polyurethanes (Gorilla glue) are waterproof and one part; again I don't know about the durability with PT material. Be interesting to look on the PT manufacturers' trade association web site and see if they have any info on laminating them--I've only done untreated.
And, of course, as somebody (or bodies) have mentioned there's always the manufactured/engineered beams but you may start talking more monies quickly for exterior application...