A good qualified tree surgeon will be able to assess it for you. We had one who took out about two ton of dead wood out of our oak, (with council permission as it has a TPO). You could hardly tell that it had been done. It retained its shape, and he said it should now be Ok for another 20 years before needing any more remedial work.
Whatever you do, make sure he is qualified, not one of the itinerant men with a chainsaw who knock on doors looking for work.
You are very fortunate to have a mature Beech tree in your garden. With lots of beech mast, you may be very lucky and have Bramblings feeding on it in the winter. I agree with the other comments; sounds like a perfectly healthy Beech tree to me.
Thank you to you all. We just want to tidy it up, that's all. I think the neighbour wants rid if it "mess" etc. seem to be his issue; we have large gardens in this area, but some people seem to prefer 'sterile' gardens & don't appreciate the beauty of such magnificent plants. I read a book review this morning in 'The Sunday Times' entitled 'The Long, Long Life of Trees'; given my neighbour's attitude, none of them would last very long!
His qualifications consist of trying to get one over on others and then basking in what he deems as his success. I wish he would move into the middle of a quarry; that would suit him.