Using an app purchased on home Apple ID on Mac using a work Apple ID I want to purchase Final Cut Pro on my home computer Apple ID, but then use it on my work computer, which is normally logged into a separate work only Apple ID.  

How can I do that?
At minimum, you need to consult with your boss. Together you should decide if you should contact you employer's lawyers, Apple's legal department or both.  If you do not follow the rules, you put yourself and your employer at risk. I found the following quote on the macOS appstore:


  You can install apps on every Mac authorized for your personal use


https://www.apple.com/ca/osx/apps/app-store/

Read the legalese:
https://www.apple.com/legal/

You could contact apple legal.
https://www.apple.com/legal/contact/

You would need to logout from your work ID on your work computer and log back in with your personal ID.  You would download and install the software.  You would log out of your personal Apple ID then log back into your corporate Apple ID. To get updates, you would log out, log in and log out.