What exactly are the reasons that you need to produce Texinfo from
some other source language, rather than maintaining them in Texinfo?
I don't know about doxygen, but javadoc has a well-developed mechanism
for maintaining the documentation for functions/classes/etc. in the
source file, and standard tools for extracting it. I didn't know there
was a way to turn that into Texinfo, but if there is, great, I think
it's worth supporting.
As for the original problem, maybe automake could have a variable
BUILT_DOCS which lists texinfo files which are created, analogous to
BUILT_SOURCES. Just an idea, not claiming it solves everything or is
easy to do ...
