In my personal case and since we have a tolerant code for design of columns built of 2 Cs with battens, I would use the whole member with N=0 if for flexure (I have a mathcad sheet at least for this, maybe 2). Hence I would only need to acknowledge external bracing, akin to that of ends.
Your view I see at least critcable in some aspects. 1) Not stating a maximum distance between battens, you can put 2 or 60 battens and obviously to the component chords it won't be the same; the specification of maximum separations for columns ensure the individual chords won't buckle ever before the built-up member. Then one member braced to another of equal stiffenss (and once joined so will behave) is within the reach of parallel sidewise buckling, obviously for a Lb bigger than the distance between battens. The main conceptual failure here is that the bracing forces even if weak need be effectively passed fo fixed points. Anyway relative bracing (which yours is) is used in the cross braced sections in bridges made of parallel stringers. Just 2 chords or stringers can make the likelihood of the initial imperfections be in the same direction and so I don't recommend it.
Furthermore the required stiffness in any bracing is that at the brace point.
In any case, the built-up member can be checked as a beam with the continuous bracing as per Galambos, but again the effective shear modulus of the continuous restraint is needed. Discrete bracing would have clear distance to use and I would make it to coincide with a batten.