I have no experience of beech trees. But I do have some big oak trees, which probably cause a similar sort of problem.
Personally, I think that big trees should be left alone, unless there's a compelling safety reason. You say that the tree is 'lovely'. When a large tree is cut back, then that does inevitably spoil the appearance of the tree. I have seen some pruned trees which look really hideous.
Large trees only grow relatively slowly, so it's unlikey to get much bigger in just a few years. There's no hurry to do anything immediately.
You should be able to determine the age of the tree by measuring the girth (circumference at chest height). An oak tree with a trunk 1 foot in diameter (3 feet circumference) would be 50 years old. 2 feet in diameter=100 years old, etc.