It's not really that difficult. First, you compare the free parameters in the theory used to explain the acceleration.Question: do you think that one model is "more likely", "more reasonable", "simpler" in the sense of Ockhams razor than the other? How do you calculate and compare the probability of a cosmological constant having a certain value with the probability of sitting near the center of a huge void?
The cosmological constant has one free parameter.
The void model has three free parameters (our position in three spatial dimensions).
Things aren't looking so good for the void model already.
To do this in more detail, it makes sense to compare the fraction of parameter space that is consistent with the model to the entire parameter space. With the void model, this requires some estimate of the frequency of such nearly-spherical large voids, but we can easily provide an extremely pessimistic estimate by taking the maximum value of this frequency that is still consistent with the void model explaining acceleration. We then compare the number of galaxies that lie close enough to the center of such a void to explain the observed acceleration to the total number of galaxies. This gives a rough estimate of how likely the model is.
To contrast this, we can compare the current error bars on the cosmological constant to the available parameter space for the cosmological constant that is consistent with an old universe that forms galaxies.
Now, I haven't done the numbers here, but I'd be willing to bet that the void model will end up with a much, much lower likelihood than the cosmological constant.
Bear in mind that the numerator of this fraction that makes up the likelihood depends upon experimental precision, so we can't take the likelihood itself as being physical, just a means of comparing between different models.
Using this sort of analysis, it is not at all difficult to compare models even when the experimental support for two competing models is identical. The result may, in some cases, be ambiguous, but it's still possible to do the comparison, and certainly not outside of the realm of science.