Can I publish the documentation under FDL v1.3 but the image under another license? Case yes, what would be a common license for this purpose?
IMHO, the GFDL is a rather complex and inflexible license, but this is your choice: if you understand it fully, go for it. My brain hurts when I read the invariants parts.
That said, you can license your image under another license or dual license it under the GFDL and another license. This is your stuff, you can do whatever you want with it.
I would suggest to look at the Creative Commons licenses for documentation and images in general.
Would it be possible for me to "detach" the image licensing from the license of the rest of the documentation?
Yes. This is a common case in Linux distros (and in Mozilla too I think). For instance, as far as I know, RedHat logos included in their Linux distro are under a different license than the rest of the distro.
Which means that when someone creates a derivative distro from RHEL (such as CentOS or Oracle Linux) they cannot reuse the image branding in their own distro and need to replace this first. Ubuntu does the same.
Now again, my advice is to keep things as simple as possible. The more complex your licensing, the harder you make it for you potential users to consider using or contributing to your project.