The sounds are called from block 10 - NiTextKeyExtraData. If you expand the keys, you'll see at what times the sounds are called. I don't know enough to tell you if removing that entire branch will remove the sound and not bugger up the nif. You could try removing the block/branch or just removing the text from the keys and then saving as a new nif and then pointing the Geck object to your new nif and see if it works or not.
I deleted the strings pointing to the sound and it seems to work fine - now the sign is completely silent. Of course, now
all Nuka Cola Signs will be silent so I might give the one in Trophies a unique ID pointing to the altered nif at some point. Thanks for the help - much appreciated!