The approach depends on the purpose of this process. I assume you are using RNase H to remove short RNAs which remain bound at the promoter. Are you removing the abortive transcripts in an attempt to modulate T7 efficiency, or are you trying to analyze the RNAs which remain bound?
If you are looking to remove the Gs from the 5' end of your full-length transcripts to study 5' UTR influence, I don't think RNase H will be appropriate for your application.