Is there a way to access the Properties object of external documents?
No. Properties cannot be shared between scripts.ref A script can only modify its own properties (be they ScriptProperties, UserProperties or DocumentProperties).
However, you can expose a scripts Properties via functions - for example, a library could provide an API to multiple scripts so that they could read & write a set of common properties in the library. See How to pass parameter(s) to timed trigger function in library script for more about that.
You could adapt that to your situation by using the library's properties as a communication between your master script and the slave spreadsheet scripts, or you could use slave sheet IDs as keys to properties for slave sheets that are stored in the library's properties.
The question you've posed is an example of an XY Problem, so it's possible that your focus on PropertiesService is precluding solving your actual problem. If that's the case, you may wish to ask another question that focuses on the problem, rather than a possible solution.