I would definitely take this approach. I know I have had juices which have sat in my fridge for a few months start a fermentation. And they would not have been "All Natural / No preservatives."
I think a lot of products like this have enough preservatives to extend the shelf life quite a while, but not stop fermentation for ever.
As Nighthawk points out, your assumption makes complete sense, but I would not attempt it on a large batch without some sort of trial run.
Assuming sorbate is listed as an ingredient, the question would be how much? You could always add an extra shot of your own to be safe.