As is always the case in physics, there is no proof.
But if your scenario were true, it would have to be rather fine-tuned in order to create the observed expansion of the Universe.
First of all, the expansion is observed to be highly isotropic, i.e. the expansion rate is the same in all directions. Hence, your lumps couldn't really look like your drawing, but would have to lie in a shell around our Universe.
Second, the matter farthest from us would achieve a larger acceleration since it were closer your surrounding matter. In fact the opposite is observed; in the "local" Universe, the expansion accelerates, whereas in the distant Universe (which due to the finite speed of light also means the early Universe), the expansion actually decelerated.
Third, we would have to be located near the center of the Universe which, while not impossible, would be regarded as a highly improbable coincidence.
Fourth, preferably you would have to come up with some mechanism that could result in such a configuration of matter.
Dark energy, at least in the form of the cosmological constant which is usually assumed, is so far (!) the simplest explanation for the observed fact that the expansion of the Universe is accelerating, and that the geometry of space seems to be flat. But several other models that fit the observations exist, and although I personally think dark energy makes the most sense — partly because this is so far from my field that I'll have to rely on what I read — I think most cosmologists are rather open to the possibility that we may one day have to severely reshape or even reject this model.