Congrats on a really good story this is one of the best HP/DC crossover stories I've ever seen.
My only dissapointment is that it's going to be a Harry/Kara story as I'm a big Hermione Fan and would like to see her with him even if she wasn't his only partner. [I even thought of a reason why you might have considered a multi-ship for the story]
If all of the powers of Oa were transfered to and into Harry it seemed plausable to me that that power might be passed on to some degree to his children. Having several wives would mean he could have more children/grandchildren/etc... and could within his own lifetime build up the Potter Family into a powerful force for Justice.
But even without any H/HR romance I like the story and will stick with it. I wonder if you have plans to send anymore of the Green Lanterns traditional enemies to Earth to try and destroy Harry. Either that or having some problem arise in some part of the Galaxy which only a Lantern can solve.