Spot on, you should fight.
Just a heads up, it is very rare (if they do at all) that they will change the details (at least I have not seen a case), so you should think of your alternative at the same time.
I don't think another IELTS is necessary, rather you can have a notarized translation of your original NID with old spelling, plus an affidavit that you have changed your name spelling from this to that should be enough. DIBP might ask questions, but you have your valid cause.
Having said that, there is always a risk that they won't accept your reasoning (though I think the chance of this happen is slim). So, you can as well reappear in IELTS if you want. Also, if I remember correctly, you have 60 points, so it might be worth to try a second time to increase your IELTS score (a all 6 to all 7 in IELTS will give you extra 10 points, which is a big deal).