Hardness of computing circles with max number of lattice points
It's probablyly much easier than factoring $n$, because you're looking for smooth numbers smaller than $n$ rather than for difficult factorizations; a smooth number is one that has many small factors, and these are the distances which will give you the most points on the lattice. If the center has to be at $(0,0)$, then you calculate how many lattice points are at distance $5^{a/2}13^{b/2}17^{c/2}29^{d/2}37^{e/2} \ldots$ using the formula for the number of ways of expressing a number as the sum of two squares. You can do the same thing for the circle centered at $(0,\frac{1}{2})$ or $(\frac{1}{2},\frac{1}{2})$. You can then do an optimization to find the largest value where this distance is less than $n/2$. I expect that this will be possible in polynomial time using dynamic programming.

I cannot imagine that the largest value would not correspond to the center having coordinates other than $0$ or $\frac{1}{2}$, but this should be proved.