If material has to be removed, you have to cover that area with the mill. If you only want to remove some of the surroundings and the pockets in the claw, you can draw another vector around the part you want to remove.
Say, you want about 10mm around the claw removed, you can offset the claw's contour by 10mm and use that vector to select the part you want to remove, rather than the material-outline..
It's just a matter of selecting the part you want to remove.
If it's getting to complicated, don't be afraid to generate 2 or more toolpaths for it.. Your computer and machine will not notice it..
Kees