Is there a way to continue using the Mac Photos book editor to create a photo book then use third party service to print?

Apple no longer provides a Photo Book service so you can no longer create a photo book with it. If you already have created a photo book with the Apple Photo Book service before it was discontinued you can open it in the sidebar and convert the project to another app that will accept it. Or you can print out a PDF of your old photo book project and send it to another service for printing. I believe Mimeo will do that.

However, you can from within the Photos app, click on File/Create/Book/AppStore to get a list of photo book apps

that you can download and use as extensions to create photo books. Then when you click on File/Create/Book the downloaded photo book app will show as an available extension that you can click on. Many people use Mimeo or Motif, but there are

several other choices.

-- Rich

Here's a list of available printing services available thru Photos: Post 9/30/2018 Printing Options for Photos.

If you are able to create a PDFfile for that book there's a 3rd party book provider which used to do the printing for Apple as a sub-contractor, Mimeo, that will take PDF files created by iPhoto and Photos and print and bind books and calendars. 

Also you can upload the PDF file to PrestoPhotos which can also print and bind books and calendars from iPhoto or Photos pdf files.  They've been doing it for some time now.  

Use the PrestoPhotos or Mimeo links above for direct access to the PDF file uploading and book/calendar ordering page for each site. 

If you can't create a Pdf of the book then the two services above do offer apps that work with Photo for making books.

If you want to be able to create photo books in Photos, you must not upgrade beyond macOS 10.13 High Sierra with Photos 3.0. The tools to create an Apple photo book are no longer available in the newer versions of Photos. If you create a photobook in Photos 3.0 you can save it as a PDF and submit it to Mimeo Photos or Presto Photos for printing.

• Mimeo used to print for Apple and can handle PDFs created in Photos, see: https://import.mimeophotos.com
• Also Presto Photo can print from Apple https://www.prestophoto.com
• If you are not located in the US, there may be local printing services, which would be more economic with respect to international shipping fees and shipping time. In Germany we can use photographerbook to print existing Apple PDF files: https://www.w2p-photographerbook.de/fotobuch-aus-aperture-iphoto-fotos

But being stuck with an older system version and older Photos version would be a high price to pay, just for the ability to create books. Photos 4.0 on Mojave is worth the upgrade. IMHO, it is the best version of Photos so far.