We have
$$ \left|\sin x\right| = \frac{2}{\pi}-\frac{4}{\pi}\sum_{m\geq 1}\frac{\cos(2mx)}{4m^2-1} $$
hence
$$\begin{eqnarray*} \sum_{k=0}^{2n}(-1)^k\left|\sin k\right| &=& \frac{2}{\pi}-\frac{4}{\pi}\sum_{m\geq 1}\frac{1}{4m^2-1}\sum_{k=0}^{2n}(-1)^k \cos(2mk)\\&=&\frac{2}{\pi}-\frac{2}{\pi}\sum_{m\geq 1}\frac{1}{4m^2-1}\cdot\frac{\cos(m(4n+1))+\cos(mn)}{\cos m}\end{eqnarray*}$$
and *if* the series $\sum_{m\geq 1}\frac{1}{m^2\left|\cos m\right|}$ were convergent, your $\limsup$ would be finite. This is not the case, since $\left|\cos m\right|$ can be as small as $\frac{1}{m^2}$ if $m$ is constructed from the numerator of a convergent of $\pi$ (consider $m=573204,m=52174$ or just $m=11$ from the Archimedean approximation). And if $\cos m$ is very close to zero and $n$ is even then
$$ \cos(m(4n+1))+\cos(mn) = T_{4n+1}(\cos m)+T_{n}(\cos m) $$
might be dangerously close to $+1$ or $-1$. Indeed the convergence of the involved series follows from a pretty heavy machinery, relying on the fact that the irrationality measure of $\pi$ is finite and we have the theorems of Denjoy-Koksma, Erdos-Turan and Van der Corput for exponential sums (see the brilliant answer of i707107 here. I agree with him that the application of the EMC formula to a non-differentiable function is very fishy. I am less skeptical about standard tools in harmonic analysis and maximal operators.) Besides that, the computation of the exact value of the wanted $\limsup$ is both extremely difficult and probably irrelevant. This brings to the table an Italian motto:
$$\text{"Non ora, non noi"} $$
meaning that the exact computation will be carried out by *not us, not now*.