Those longitudinal bars function as subframe connectors, as long as the hoops are secured to the chassis.
Whatever you decide on, don't go crazy on wall thickness. Even if you do 2x2, 0.125 wall is as big as I'd go.
One other thing, all steel has the same spring constant, so even the lowliest piece of 1019 hot rolled will have the same stiffness as a piece of 4340. The low carbon steel will be much easier to weld, so don't get fancy.
