Or you could just laminate that cardboard with some fiberglass and epoxy, on both sides. A roll of 5cm x 3000 cm fiberglass woven cloth is $9 incl shipping on ebay. Buy some two component epoxy at the local hardware store, like 10 fl. oz. for $15.

Cut your cardboard and fiberglass to whatever size and shape you like. You're going to make a sandwich: fiberglass, cardboard, fiberglass. Wear disposable gloves. Cover your benchtop in two layers of cling film. Put your sandwich down, carefully rub epoxy into the top fiberglass, two layers shrink wrap on top, flip the thing over, rub epoxy into the second fiberglass, more cling film. Put it in press with some weight, leave overnight. If your cardboard has some thickness and good compressive strength, the strength of the resulting sandwich is probably such that you won't be able to break a 10 cm long stick with your bare hands.

3D printing is cool. But for random repairs around the house, or building strong flat things like this, epoxy and fiberglass is the bee's knees. We've built stand up paddle boards with this technique, just replacing the cardboard with XPS plates. No vacuum bagging required, really. (Of course it would give further strength and much improved aesthetics.)