Don't rule out the free online version of Microsoft 365 by signing into a created Microsoft account. Then you can generate a Word .docx or PDF from there for the commercial printer. No one accepts native Pages documents.
If the commercial printer is expecting a proper PDF/A or PDF/X standard document, it cannot be generated by Apple's PDF frameworks. One can tell Apple's Preview to do that but it is not a validated PDF/A document. Usually, these commercial printers stipulate what specific source of documents they require.
Pages only generates PDF v1.3 documents and if the printer wants newer (e.g. v1.5, 1.7) PDF documents for features exclusive to those newer versions, then they will reject the submitted PDF from Pages.
PDFs created by Apple's Pages have the PDF Metadata Creator set to Pages and the Producer set to Apple's Quartz PDFContext. No hiding that fact from a commercial printer.
On the subject of Word documents, Pages exports Word documents that are a translation from the Pages content. This will not have all of the styling or internal document contents that MS Word (or LibreOffice) would write into Word documents saved from these applications. Apple omits the saving application name, and MS Word and LibreOffice write "Microsoft Office Word" and "LibreOffice/24.8.4.2$MacOSX_AARCH64 LibreOffice_project" respectively. LibreOffice saves yet a different internal Word document format than does Pages or MS Word though MS Word can open either.