I'd like to make an NPC character who, whenever you talk to them, a nearby invisible creature with their name forcegreets and talks to you instead, to cut around disposition, erroneous non-filtered NPC dialogue, etc.

But I'd also like this NPC to have companion share functionality. Is it possible for me to somehow open the companion share window with the NPC without opening dialogue with them?

I'm not sure what the bonus would be to have a ForceGreet from the creature (surely annoying if you DO want to talk to the companion?). In Morrowind, there's no spoken dialogue so there's no time delay when you open the dialogue option. You can just have a short, even blank greeting (just a . ?) and then click straight on the companion share option.