You should have asked here — before writing your booklet in Pages — if Pages should be used for professional print production. The answer, for years, has been no — because this is not what Pages was designed to provide.
Pages cannot produce "print ready PDF" that a professional print shop requires. This is a limitation of both Pages mucking with the color model, and not providing crop and bleed adjustment capability, or converting fonts to outlines — as much as the Apple PDF library that cannot generate PDF/A or PDF/X PDF standard documents upon Pages export to PDF.
This so-called "professional" print shop should have provided you with a list of just what "print ready PDF" means to them as requirements, or at minimum, their website should provide this information.
I was going to recommend Swift Publisher 5 until I downloaded and took a closer look at it. Although it gives you more control over CMYK color model, and offers crop and bleed settings, as well as convert fonts to outlines, it still uses Apple's PDF library to export to PDF, and that can be a problem for print shops that require PDF/A or PDF/X documents. One of the Swift Publisher's PDF formats is PDF/X3, but I don't know how that is going to actually work without a private PDF library. Swift Publisher does not import other document formats either.
Affinity Publisher ($50USD - Mac App Store) has its own professional PDF library, and all the capabilities you would need to meet, or exceed any print shop PDF requirements. It too does not import word processing documents, and you would have to start from design scratch, including the learning curve. There is a free trial from the Affinity site and training videos.