It seems that some site-specific cleavage is possible (Nucl. Acids Res. (1979) 7 (1): 179-192.doi: 10.1093/nar/7.1.179) - but in this paper they were using tetramers to target the enzyme and you only have 3Gs.
On the other hand, I'm pretty sure you don't actually need GGG for the T7 transcription to work. You absolutely need atleast one G and transcription from GC or GT won't be very efficient. GG or GA would be pretty good. When you have GGG, you can actually get template slippage such that your RNAs will have 1, 2 , or 3 Gs (I believe) which might make an RNaseH strategy even more complicated.
Does your experiment absolutely require a certain 5' end? If so, I might actually suggest adding more sequence between the GGG at the beginning and your region of interest so it is easier to target with something like RNaseH and a specific oligo or a DNAzyme AND so you can resolve cleaved from uncleaved.