I lack time to give you the details, but the general formula is
$$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$
More generally, for a block-circulant matrix with $n$ square blocks $A_0,\ldots,A_{n-1}$, the formula is
$$\det K=\prod_{\omega^n=1}\det(A_0+\omega A_1+\ldots+\omega^{n-1}A_{n-1}).$$
At least, you see easily by linear combinations that if some of the factors above vanishes, then $\det K$ vanishes too.

Perhaps I'll write details tomorrow morning.