Exact complexity of a problem in $\cap_{m \geq 2}\mathsf{AC}^0[m]$
I do not think this is in AC0 and I can show a lower bound for the related promise problem of distinguishing between $\sum x_i = 0$ and $\sum x_i = 2$, when $x \in \{-1, 1\}^n$. Similar Fourier techniques should apply to your problem, but I have not verified that. Or maybe there is a simple reduction. 

Suppose there is a size $s$ depth $d$ circuit that computes a function $f: \{-1, 1\}^n \rightarrow \{0, 1\}$ such that  $f(x) = \sum_i x_i$ whenever $\sum_i x_i \in \{0,2\}$. Because for a random $x$, the probability that $\sum_i x_i = 0$ is $2^{-n} {n \choose n/2} \approx n^{-1/2}$, and for each such $x$ there are $n/2$ coordinates that change the value of $f$, the total influence of $f$ is $\Omega(n^{1/2})$, which is roughly the same as majority (because you included most of majority's sensitive inputs). By a theorem of Hastad (see Colorraly 2.5 in Ryan O'Donnel's notes), this implies that 

$$
s \geq 2^{\Omega(n^{1/(2d-2)})}.
$$