I wrote something like this using reqwest for a Waveshare 7.5inch display. Also includes a dithering step so all that is needed is a web server serving an image. No conditional requests though but should be easy to add. It is fully statically compiled (even uses musl and rustls), runs on a very bare-bones Linux environment on a Raspberry.

https://git.pixelspark.nl/pixelspark/sparkboard

If it's really just the screen refresh that's to be avoided, I think just comparing old download /new download with "diff -q" would be a reasonable solution. Of course, perhaps the Kindle doesn't have a diff binary. But maybe it has "sum" or "md5sum" or something else in that space.