How come we've assumed the issue has something to do with the RF connection and not considered that the Famicom is defective? If you're willing to modify it, just give it an A/V mod and forget about RF. It might be possible to get around whatever may possibly be wrong with the RF out, too. At the very least you could verify if sound output is working in the first place...

As for RF boxes potentially adding interference, I wouldn't know if they can or not, though I suppose they could. Never thought of it actually.