Tarev,
Thanks so much for your comprehensive reply.
• I will water from the top now -- I see how that makes sense.
• I'm encouraged by what you say about watering: I've certainly let the medium dry out before re-watering. if anything, it sounds as if i might have been a bit too sparing with the water, for this tropical cactus.
• Your info suggests re-potting in a shallower, wider tray or dish rather than a traditional pot. Maybe the roots would then spread laterally to provide the support that's missing just now; and I'll stake for the moment while the roots are lengthening.
I might also try to adjust the medium. At the moment, it's contains a lot of perlite. If my cactus is tropical, I might reduce that proportion in favor of dirt and sand, The problem with the perlite-rich medium is that it's crumbly, so the roots can't bind easily to provide the anchoring and support necessary to keep the plant upright.
No plans at the moment to re-graft -- the graft is the bright spot, and is holding fast!
Thanks so much,