Hello, pseudoelvis.
First of all, thank you for using all of the tools and running the tests  

I just updated my Edge installation and then tested this on my system with all of the same versions you have.  It works for me.
It was a good idea to try and reinstall.  But to do that you must uninstall first.
• Make Edge the active app.
• Click in menu: Microsoft Edge > Microsoft Edge Extensions.
• At the bottom of the 
BookMacster Sync card, click 
Remove and affirm your intention.
Now you can re-activate BookMacster and the 
Install button should work.
Some background information: In addition to prompting you to install the extension into Edge, that button also installs a so-called 
Native Messaging Host file which tells the extension how to find BookMacster and vice versa.  It is located at this path in your Home folder:   
~/Library/Application Support/Microsoft Edge/NativeMessagingHosts/com.sheepsystems.chromessenger.jsonProbably that file is missing or is containing outdated information, and reinstalling will fix it.