Is it possible to use `tmutil delete` or another command to delete one file from all backup intervals?

In some sense, you are right, but as noted in the comments you need to be a bit careful. I am not really sure about references, but I'll reformulate Gitler's construction for you:

Take $G$ a group. The first thing is to take the classifying space $K(G,q)$. This can be done functorially, see this MO-discussion. (Unfortunately, the best way to do this is via realization from simplicial sets - I do not know a nice way which does everything strictly with CW-complexes only.) For $\pi$ acting on $G$, this provides a representation of $\pi$ on $K(G,q)$ (or better, as noted in Qiaochu Yuan's comment - on the model of $K(G,q)$ as realization of some explicitly given simplicial set). The generalized Eilenberg-Mac Lane-space $L_{\pi}(G,q)$ is then defined as the Borel construction $E\pi\stackrel{\pi}{\times}K(G,q)$, where $E\pi$ is now any contractible CW-complex with a free action of $\pi$, and $\stackrel{\pi}{\times}$ means taking the product and dividing out the diagonal $\pi$-action. This is the construction in Gitler's paper.

It is then a general fact in equivariant cohomology, associated fiber bundles etc. that there is a fiber bundle $K(G,q)\to L_{\pi}(G,q)\to K(\pi,1)$ as written in the question.