Manage "Document Type" column for each library which have different options

We have this case,:-

1) A site with 10 document libraries

2) For each document library we need to tag the documents with a managed metadata column named "Document Type"

3) Use the PnP Modern search web part, to be able to filter and search the documents using the "Document Type"

I found 2 approaches to follow:-

1) Create a single managed metadata site column named "Document Type" >> with all the options and user will select the related term. 

*Advantage:-

 - Simple

 - Single managed metadata property to be used inside the PnP Modern search web part

*Disadvantage:-

 - Users will get all the options even the ones not related to the current document library, when tagging the documents inside the document libraries.

2) Create 10 columns named "<<document library>>Document Type". each column will only contain the related the related options for the current document library

*Advantage:-

 - Users will only get the options related to the current document library

*Disadvantage:-

 - not sure how we can filter the documents from all the libraries using a single column? For example, can i configure a single RefinableString  managed property which have the 10 underlying crawl properties? so i will have a single managed property that span the 10 crawl properties and which i can use inside the PnP Modern search web part? will this work?