How about generating latex source code, compiling it and getting the page count of the generated PDF? Reorder your set of questions and see if the result is better or worse. Optionally do it in a smart way to reduce the number of PDF compilations you have to do. (Simulated annealing comes to mind for example.)
I think it would be easier to find a library to find the last line on a PDF page than it is to parse unzipped odt files and basically write a layout engine that does the same as libre office just to get the number of pages.
Maybe you can even get Tex to put it in the log during compilation. That would be the most convenient option and seems reasonable to achieve.