If you can't accept having part of the previous chapter on the same page as the start of the next chapter, then the problem devolves into treating each page as an image, and doing image editing until it looks to your satisfaction, then PDFing the images together.
PDFs are not "reflowable", meaning, if you delete some text, the rest of the text won't automatically "move up" like it would in Microsoft Word or other page-based document editors. PDF simply does not work that way, and there is absolutely no way to get it to behave that way and retain 100% fidelity of the input content.
You could use Adobe Acrobat to attempt to "save as" the document into Word format, but this is an extremely imperfect approximation that attempts (sometimes poorly) to define the PDF's contents in terms of MS Word. The resulting document can be very difficult to modify, and may not look anything like the input document, especially once you play with it a little bit.
It also depends on how the input PDF was generated. If it was generated from images, then all the text would have to be OCR'ed to do any modification of it anyway, and OCR has its own problems (extremely inaccurate unless it already knows the font being OCRed and the DPI is very high). If the PDF contains the actual text, the problem gets a little easier, but if you want to retain 100% of the original formatting, graphics, etc., the problem again becomes extremely hard.
I would just live with having part of the previous chapter on the same page, and use one of many tools that exist to extract the pages you want from the PDF. You can use for example
pdfedit to extract the pages and insert them into another PDF on Linux. It is very easy to slice out pages from one PDF and put them back into another PDF at an arbitrary place while maintaining 100% fidelity of the content. It's when you want to change the contents within a page that it becomes nearly impossible to do so without losing fidelity or spending a huge amount of time and effort manually editing rasterized images of the PDF pages.