A quick numerical investigation for $N$ up to 2048 leads to a conjectured expansion of the largest eigenvalue, 
$$
\lambda_0(N) = 4 - \frac{2\pi}{N} + \frac{\pi^2}{2 N^2} - \frac{\pi^3}{24 N^3} + \frac{\pi^4}{48 N^4}
- \frac{\pi^5}{c_5 \, N^5} + \ldots.
$$
The coefficient $c_5 \approx -114.63(1)$ is negative and seems to be a more complicated expression.
I don't think that the problem simplifies for $N$ being a power of two.

For reference, I'll append some high-precision results:
\begin{array}{lll}
N & λ_0(N)& c_5(N)\\
 128 & 3.95121320281088603898478521135 & -112.33736106154642063 \\
 256 & 3.97553152996970070083424756460 & -113.483268010223740 \\
 384 & 3.98367098180919839228844082991 & -113.86471637351927 \\
 512 & 3.98774696887736782094115794509 & -114.05534610371244 
\end{array}