(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly perturbed in the manner described below.)

Imagine an arbitrarily large two-dimensional rectangular lattice with vertices $(v_1, v_2, ...) \in V$ and edges $(e_1, e_2, ...) \in E$, where the edge-length in the lattice is $L$.

Place a point, $Q$, somewhere internal to the lattice (and sufficiently far away from the lattice periphery). Simulate Gaussian noise about $Q$ by sampling $N$ points $(p_1, p_2, ..., p_N) \in P$ according to the PDF of a Gaussian centered on $Q$ with defined values for $\sigma_x$ and $\sigma_y$ that are some fraction of the lattice edge length $L$ (let's simplify things by setting $\sigma_x = \sigma_y$). Bin these points to the nearest lattice vertex, and generate a set of counts $(c_1, c_2, ...) \in C$, where $c_i$ represents the number of points binned to the vertex $v_i$.

Now, define a set of real numbered values over the interval $[0,1]$: $(k_1,...,k_N) \in K$, where $k_i$ values are drawn from a Gaussian distribution with a mean $u_{y}$ and a variance $\sigma_{y}$. Generate a new set of vertex counts, $C^*$, by multiplying each $c_i \in C$ with the corresponding $k_i \in K$: $(c_1*k_1, c_2*k_2, ..., c_N*k_N) \in C^*$.

As a function of $u_{y}$ and $\sigma_{y}$, what will my worst-case error be in terms of determine the true position of $Q$ provided the counts $C^*$?