I think that you can do that

I saw previous post regarding similar situtation. That guy moved to Texas district after he submitted NIW at Nebraska center . Later he submitted another application to TSC and the case was approved while the one in Nebraska was pending.
He mentioned in his post that he just used photocopy of the recommendation letter and submitted everything identical to his previous submission. He said that he also included a statement saying that the photocopy is identical to the original one. As far as I remember, he said that he didn't put anything newer there or mention his previous submission in the new submission.
Also my lawyer talked to me about the possibility of submitting my NIW to another center if this one failed. He said that it is fine to have two application in different place. So I really don't think that there would be any problem if you want to have second NIW. But I am not professional and it is just my opinion.
Good luck. Hope that help.