How are you getting PDFs from Gutenberg on a regular basis? The books I've looked at have had:
EPUB3 (also for Send to Kindle)
EPUB (no images)
Kindle (delivers -kf8.mobi so rename to .azw3)
Older Kindles (delivers .mobi)
Plain text (UTF-8)
HTML (in a .zip file)
I just did a random check of 20 books and nary a .pdf to be found. According to Gutenberg's documentation, "Project Gutenberg has some PDF files that were created from other markup (TeX, LaTeX or others), usually for math-heavy eBooks."
As for your Paperwhite only reading mobi format? As far as I am aware, any Paperwhite can read mobi, KF8/azw3 and KFX for Amazon formats plus PDF and TXT. I downloaded the .mobi and-kf8.mobi download of
Little Fuzzy (pg18137), rename the -kf8.mobi version to .azw3 and copied them to my PW4 and no issues with displaying them.
I checked Gutenberg for the USA, Canada and Australia.