I figured it out. If someone else encounters this problem:
If you export your different LODs as single FBX files make sure you give them the same name in 3ds MAX. I gave mine names (in MAX) like SK_coat_LOD0, SK_coat_LOD1, etc. But inside MAX they all need to have the same name. You can of course name the exported FBX however you like.
Cheers,
Florian