I managed to achieve what I want. I used the
Entity View Modes (evm) module. I created a custom view mode under Configuration » Entity View Modes » File.
This view mode is now available under Structure » File types » Image » Manage file display
Here I can select the checkbox Image. And then select my desired image style under the Display settings in tab Image select box Image style.
This new view mode is now usable in display suite. If you go to the content type Gallery and click on Manage display where display suite hooks in you can now choose the formatter Rendered file for the field Gallery media. In the formatter settings you'll find your new view mode. Select it and et voilà you get the image with your desired image style.
I also found a solution with display suite itself. Without the needing for the additional Entity View Modes module.
Just create a new view mode for File using display suite. Then in the Structure > File types menu you can activate the view mode for Images. And choose your custom image style.
This view mode you can then select in the field template settings like in the description above.