Install and start the application from your hard disk?

I'm in agreement with Camelot above. This is a little confusing. If you just want to install and run a program:

1. Navigate to the App Store
2. Search for Final Cut Pro
3. Purchase/download the app.

Then navigate to Finder > Applications, and double click Final Cut Pro, then you should be able to use it whenever you want.

Ah, you have 256 GB of Storage, not Memory. :)

You can move the FCP Library to the external drive...

https://support.apple.com/en-us/HT208805

I admit I'm a little confused.

You ask :

"Would it be possible for me to install Final Cut Pro on the hard drive and then start using it all the time?"

as opposed to what? how else do you expect to run it if it's not installed on the hard drive?

You're welcome & good luck! :)