OK, after a break from flying wings to build a Mini Mustang, I'm back with "Get 'er Dunne" III. This one has the control surfaces moved to the top wing, for no particular reason other than they might be less damage prone. For the cross bracing you suggested, I used fishing line and BBQ skewers. The skewers are glued in place in the leading edge of wing. One at a time, each skewer has more hot glue applied, then the fishing line is wrapped around several times and held taut until the glue dries.
I got to maiden this today with some drone pals in 8-12mph winds. The cross bracing definitely reduces the amount of wing flutter. It still is a box kite in the wind, but the flex within the structure is noticeably less than on the previous model. After maybe half an hour of flying (which was very enjoyable in FPV), the cross bracing seemed to retain most if not all of its tension. We'll see if it lasts.
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Edit: did not try a step airfoil, maybe after I inevitably destroy this one.