Not sure if this helps but...
In iwama ryu thats morotedori kokyuho. Looks a lot like a Sokumen iriminage in other styles, but it is quite different in how it is applied.
Well, as far as I'm concerned, it's kokyu-ho if you do it as an exercise, and kokyu-nage if you finish it as a throw. I have done it from morote-dori, ai-hanmi and gyakuhanmi katatedori, ryotedori, shomen-uchi, yokomen uchi, ryo-mune-tori, ushiro-tori, ushiro-kata tori, tsuki, ken-tori, jo-tori, and tanto-tori. Forgive me if I forgot any, but I don't think this is limited to morotedori.
As for whether it could be an irimi-nage, I don't see why not. There is a lot of irimi in it, and it's just a name after all.