Hello @Jorik Kloppenburg
If you don't mind testing an add-on, Elements Copy & Sync lets you do exactly that
Full disclosure, I'm the Product Manager of the app.
With our fields mapping, you can copy anything you want from an issue to its copy, including the Epic link.
As a bonus, you can keep this Epic synchronized, so if you change it in the original issue it can change automatically in the copied issue.
The app is free for 30 days, (and stays free if your instance is under 10 users), so feel free to give it a try, and don't hesitate to tell reach our Support if you have any question.