One of my pieces is also cracked so I've been pondering this problem. Fortunately my crack it isn't too bad.
Anyway, as an alternative to the welding/solder gun technique, I wonder if anybody has tried plastic model cement, aka 'airplane glue.' These solid molded pieces are made of ABS, a close cousin to the polystyrene used for plastic models. The trick would be to use such a tiny amount that a bead isn't formed on the front when pieces are squeezed together. Or perhaps clamping the parts and running a bead on the back side. The cement will soften and partially dissolve the surface of the plastic, so swirling around the bead before it dries (with a toothpick perhaps) would strengthen the bond. Let dry at least 24 hours (or even 48) without movement.
I haven't tried this but I'm pretty confident it would work IF there isn't much stress applied to the fixed molding. When I get done with other ongoing projects I'll give it try and report back unless somebody else beats me to it.