After spending a few hours experimenting, I figured out what the problem was in getting PHP and mod_litbook [1] working in the same server—it was entirely my problem.
I re-extracted Apache from the archives, and proceeded to add my module first, see that it works properly, then adding PHP (paying very careful attention to the directions for adding it manually) and testing to see if both worked.
They did.
I then moved the new executable to the webserver and tried it.
mod_litbook [2] didn't fully work, like last time. That was a clue I should have punced upon last time.
Turns out I wasn't running the latest version of mod_litbook on the primary webserver. The version I released was a bullet proof version that Mark [3] had (at my urging) criticized for some potential bugs that I fixed. In the process I had included a few more configuration directives and it was those missing directives that was causing my module to fail.
Ooops.
[2] https://github.com/spc476/mod_litbook