Do you still bale with wire over there? I've wondered this in the past because of the apparently high incidence of hardware disease that you appear to have. My farming life started in 1972 and even back then wire had long gone from hay bales, so much so that I've never seen hay baled with wire.
That aside, no, cows cannot eat the twine irrespective of whether it's plastic or sisal and hay should never be fed out with it still intact. If a cow ingests it, they chew it up into a ball and if you (and she) are lucky, it will be expelled at the other end. If not, it compacts in the rumen and causes big problems for the cow that usually ends up as a big hole in the ground.
Apart from the health problems involved in eating twine, nor is feeding out a bale in a lump good feeding out practice and results in wastage as cows fight to get at it, trample all over it, pee on it, crap on it and half of it never gets eaten. If feeding out in a paddock, break the bale up into it's biscuits and spread them out every 4-6 feet. The cows will clean it up and you get the benefit of spreading the grass seed around your paddock.
Welshmom, baleing twin is invaluable on a farm for all the reasons you mention. Somewhere, hidden on this computer

, I have a pattern for weaving hay sacks out of baleing twine. These are handy things to feed hay to sheep, calves and goats and prevent wastage. They can be hung off a walll or fence and the animals pull the hay through the mesh as they need it.
Cheers,
Ronnie