I checked and I have a similar problem. It just never came up because my headless actors were hardly moving. However, I did find that if any head movement is eliminated you are probably OK. The head is still there, although made invisible, so it does have an effect on the prop when it's moving.
What you could try is this: make sure that the meaty neck piece is attached to the Neck bone. Then, after applying the animation, delete any Head motion by going into the time line. Right-click the Dying animation clip and select Remove Motion => Head. That should help.