Currently this is expected behavior. If you are saving as a PDF, the PDF is a new document, and you need Export rights to be able to generate it, and as you described you need to right click on the document > classify and protect > and add a label. The code to inherit the label is not there since it's seen as a new document. There is an open feature request for this and a related discussion on Reddit. https://www.reddit.com/r/Office365/comments/fnnwym/information_protection_pdf_label_missing/
I am happy to pass your feedback to the product team as I agree that this experience could be improved.
https://learn.microsoft.com/en-us/azure/information-protection/rms-client/protected-pdf-readers