Bird ailments are extremely difficult to diagnose, even for a Vet, let alone on a forum. I do feel however, that if eggbound you should have been able to feel the egg when putting olive oil on the cloaca. Furthermore, after three day she'd be dead by now as the same orifice is used to pass excrement and urea. Eggbinding is also a continues condition and does not go away and come back for a few minutes.
As you probably know, kakarikis are very active birds and play like kittens on the ground so eggbinding is fairly rare in healthy birds. Any bird that is not in good condition is prone to eggbinding as the muscle that palpate the oviduct are not toned.
It does sound like it is trying to pass something so maybe the bird is constipated. Perhaps a drop or two of olive oil in the other end will be benificial. The condition you described is a typical reaction of all birds that have a discomfort around the claoca. Make sure that the vent area is clean and no damaged feathers are present that could 'tickle' the bird.