Explore using the System object (see
https://wiki.mementodatabase.com/index.php?title=Memento_JavaScript_Library#Interaction_with_the_system).
I've never done it, so I can only aim you at information, but the System object in Memento is intended to enable the interaction of a script with the underlying OS to accomplish whatever can be done, and I would imagine use of the clipboard might be one of those things.
The problem with querying the Web to find out how to do this is that most JavaScript (JS) articles assume you are programming a Web site, so they use DOM (Web) objects like Document or Navigator, which don't exist in Memento. You might find something there, but I'd be surprised, because there's a job to get done -- penetrating the OS to do things -- and they'll likely use the Web (DOM or Document Object Model) objects to do it.
Read at the link I gave up top -- probably using an Intent object in the process, but I'm guessing -- then I'm guessing (again) that you'll need to do something different for each of the platforms you want to support, so query with search keys like "Android clipboard API". Substitute if you want Windows, macOS, Linux, or iOS for Android. Hopefully, I'm wrong, and there's a standard across all platforms for accessing the clipboard.
Good luck, and it would be really great if you could report back with your findings, some links, or even with your solution.