I don’t think you can change a PDF. That is the only form of the document I gave him, so that it could not be changed.
"My policy is not to give out source code, but I'd be more than happy to make any changes you might have in mind as new work, which will be substantially less expensive since I can use the source as a template."
It lets him know you're not comfortable handing over your source files but are willing to work with him in regard to keeping his costs down.
Also, in regard to editing a PDF file, if your text isn't changed to outlines, it can be opened in Illustrator and the text edited. If he doesn't have the font you used, it will substitute with a default which will also affect the design.
By he is asking for your actual program file? Like asking for a Corel draw file?
I don’t know if a professional version of a PDF program could maybe these days carry the whole project over from you given PDF to elsewhere. I do not know.
PDF files can be edited in Illustrator.
Illustrator can open PDFs.