Dead, dying or dangerous means the Council should have no issue with either its complete removal or surgery work to remove that which poses a risk to your property or persons.
Best to approach the tree officer first, before you do anything, even clearing the debris, and hopefully gain his support. Up to £20k fine if you do work without approval. If s/he proves 'difficult' and you think you have a good case get a qualified aboriculturist to see if he agrees with you and can put in a case for the removal.
seeing him this afternoon