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This is kind of a tough situation because you both have valid points... You paid for your "last month" regardless of when that occurs but then you agreed to stay on at the higher rate once your lease has ended.
With that said, I think that if this got in front of a judge, he would rule that since you voluntarily agreed to extend your tenancy and accept the new terms of the proposed rent increase, that you would get credit for the $1850 towards last month's rent, but would still have to pay the additional $150. Had you just not paid the rent at the beginning of the 12th month and used the prepaid rent, it would have been at the old rate. But when you stayed on, you are agreeing to pay the $2000 a month from that point on until your lease terminates.
I am very sorry that I don’t have better news, but please understand that I do have an ethical and professional obligation to provide customers with legally correct answers based on my knowledge and experience, even when I know the answer doesn’t make the customer happy...