As long as your library has Kindle compatible formats in it, it will.
These screenshots are from the built-in browser using the simple non-Javascript interface to the calibre-web server that is part of Calibre. You can download any file listed, but you can only open Amazon compatible ones in the Kindle reader. I use KOReader on my Oasis 2 to read EPUBs, so that's why the stock UI only shows the PDFs I have loaded and the one AZW3 I downloaded for the screenshot.
Also, I hadn't kept up on COPS and it looks like the original repo I linked is mostly abandoned, but there is an updated fork someone made to work with PHP 8:
https://github.com/mikespub-org/seblucas-cops (RE:
https://github.com/seblucas/cops/issues/530 )