I made some progress in the sense that I believe that I could reduce it to a more standard problem: Morally speaking, I have \begin{align} c_K(t)=\mathrm{Im}\log\det\left(\frac{e^{tK}-Je^{tK}J}{2}\right)=\mathrm{Im}\mathrm{Tr}\log\left(\frac{e^{tK}-Je^{tK}J}{2}\right)\,. \end{align} The tricky thing is that $\log{(e^x)}$ is only equal to $x$ in certain patches. Consider the special case, where $J$ commutes with $K$, such that $[J,K]=0$. In this case, we can simplify to find \begin{align} c_K(t)=t\,\mathrm{Im}\,\mathrm{Tr}(K) \end{align} and everything is good. However, I'm not aware if there is similar simplification for more general expressions.
Question: Is there any way to describe $c_K(t)$ more explicitly with analytic functions, rather than just defining it to incorporate the Winding number by hand?
Ok, I solved the problem. We need to use the cocycle function $\eta(M_1,M_2)$, which is defined to satisfy $\varphi(M_1M_2)=\varphi(M_1)\varphi(M_2)e^{i\eta(M_1,M_2)}$. The idea is that we write $K=u\tilde{K}u^{-1}$, such that $c_{\tilde{K}}(t)=t\mathrm{Im}\mathrm{Tr}(\tilde{K})$. This can always be found by using a transformation $u$ that brings $K$ into a standard block diagonal form with respect to $J$, i.e., both of them are block diagonal (they may not quite commute, but almost). We can then use the cocycle relation to see that $c_K(t)=c_{\tilde{K}}(t)+\eta(u,e^{K})+\eta(ue^{K},u^{-1})$. This can possibly be simplified further, but the idea should be clear.
I hope this helps somebody with a similar problem in the future...