The simplest solution is probably to use pdftotext from poppler-utils to extract the text, and then use an existing command line spell-checker (spell, ispell, aspell, etc...) to check it.
Note that if you are generating the PDF, you are probably better off checking the source of that generation, as words are liable to be bro- ken by being formatted in a PDF.
You can try Syncfusion .NET Word library (Essential DocIO) to create, read, manipulate and convert Word documents programmatically without Microsoft Word or other dependencies in your machine. It offers the Word to PDF conversion ability as well.
NuGet package required: https://help.syncfusion.com/file-formats/docio/nuget-packages-required#converting-word-...