Smoothed analysis of the Partition problem
Smoothed analysis, as I have seen it, has typically been used for analyzing optimization problems, not decision problems.  The partition problem is a decision problem, which introduces the challenge you mentioned.  Perhaps one could consider the optimization variant of the partition problem: given $n$ numbers with a sum of $2S$, find the subset whose sum is as close to $S$ as possible.  I don't know whether the smoothed complexity of that optimization problem would be interesting or illuminating, but it eliminates the problems you articulate in the question.