I'm afraid there's no good way to do that that won't do irreparable harm to the white (knock-out) text. Your tube is being printed with two blues, the lighter of which is a solid colour with "holes" in it where the tube shows through for both the white text and the areas around the black text. Your darker blue colour is screened at the transition area (with corresponding knock-outs for the text, etc.), and there's only so small the dark blue dots can get. That means that there will be a definite point where the dark blue stops, rather than just gradually petering out to nothingness.
You can get an apparently smooth gradient between the two colours, but only if the light blue colour is also screened (in other words, process printing). But that means that there will be grids of colour dots that don't line up with one another (if they did line up, then the last colour printed would obscure the previous colours), and that means that your reversed text and the text boxes around the black text will be ragged and seem to have colour fringing.
If you were printing on paper (of the right sort, like a coated stock) or on a synthetic substrate that's aimed more at high-quality, don't-touch printing rather than something that needs to be formed into toothpaste-type tubes, heat-welded and aimed at less-than-delicate consumer handling, you'd be able to use a fine-enough process screen (or even a fine-enough dark blue spot screen) to get something that looks a lot like a smooth gradient to the naked eye and allows a clean-looking knockout. The fact is, though, that you're not, so you're left with the limitations of industrial package printing processes. Or you can give up on white text (at least as a knock-out, and opaque white as a printing colour doesn't usually work well when laid over other inks, even when it can work over a dark-coloured substrate) and go for a different spot colour. (Your set-up and unit pricess will go up, though.)