Hi Inces. I don't know if I understood really your question, but I had same kind of troubles with some of my characters... The problem itself is that blender don't stores, in animations strips, things that have not to change their location, rotation and so on... But Godot don't know original position of the character (and its skeleton...) in blender scene. So it happen that everything not moved during animation when imported in Godot, take a default (and generally wrong) value. Solution it's extremely simple: in the first frame of animation, when in "pose mode", press "i" key and select "Whole Character" to store all information about all the skeleton at beginning, also for bones you don't need to move. This make possible to give Godot the whole beginning situation at start of animation.
If you would like to export only some bones, ignoring the others, I fear it is not possible, because a bone must be placed somewhere in the 3d space... I solved this problem simply thinking an initial "default" state, in AnimationTree. I mean you can export a "fake action" where your character do nothing and use it as a start point for all others. So, in Godot, it become that only some bones moves, when the others stay in same position of your default state.
Hoping can help you.