A quick numerical investigation for $N$ up to 512 leads to a conjectured exact large-$N$ expansion of the largest eigenvalue, 
$$
\lambda_0(N) = 4 - \frac{2\pi}{N} + \frac{\pi^2}{2 N^2} - \frac{\pi^3}{24 N^3} + \frac{\pi^4}{48 N^4}
- c_5 \frac{\pi^5}{N^5} + \ldots.
$$
The coefficient $c_5^{-1} = -114.63(1)$ is negative and seems to be a more complicated expression.

**Added 16.07.22, 18:00 CEST**:

Extending the high-precision calculation to $N=1024$, the next two constants seem to be $c_5=-67/(2^6 5!)$ and $c_6=653/(2^7 6!)$. These terms should be verified with a precision higher than 30 digits.

**Edited 17.07.22, 15:00 CEST:**

I did this verification with 100 digits precision, verifying $c_{5,6}$ and extending the series to 10th order.
Therefore,
$$
\lambda_0(N) = 4 - \frac{2\pi}{N} 
+ \frac{\pi^2}{2 N^2} 
- \frac{\pi^3}{24 N^3} 
+ \frac{\pi^4}{48 N^4}
+ \frac{67\pi^5}{7680 N^5} 
+ \frac{653\pi^6}{92160 N^6}
+\frac{32519 \pi ^7}{5160960 N^7}
+\frac{135001 \pi ^8}{20643840 N^8}
+\frac{45750727 \pi ^9}{5945425920 N^9}
+\frac{1198585643 \pi ^{10}}{118908518400 N^{10}}
+\ldots.
$$
Note that the denominators of $c_n$ are divisible by $n!$.

All constants were successively determined by a least-square fit of a 20th order polynomial in $\pi/N$ to $λ_0(N)$. I did not find a known function with this series expansion.

For reference, I'll append the 30 digits results:
\begin{array}{ll}
N & λ_0(N)\\
 128 & 3.95121320281088603898478521135 \\
 256 & 3.97553152996970070083424756460 \\
 384 & 3.98367098180919839228844082991 \\
 512 & 3.98774696887736782094115794509 \\
 640 & 3.99019456589827273460033342852 \\
 768 & 3.99182713285240169577614505117 \\
 896 & 3.99299366147114034625800714548 \\
 1024 & 3.99386878184121881247367446866 \\
\end{array}