Wow, I’d love a beech like that.
accept the tree, use it as a feature. A rockery, with alpines or woodland type plants, would be very nice. And if you’re wanting a path, a bark or slate chipping base with natural looking stepping stones, would look nice. Though that is more a natural looking garden, if you’re wanting a modern sort of clean lines/square edges, maybe not for you.
old beech’s do not cope well with hard pruning, you’re only meant to lightly prune if at all.
that tree was probably planted around when the houses were built, given you said they were 60s houses. A beech around 10 years old tend to be around 4 metres tall. So this is quite an old tree. And if on a housing estate will be a shorter beech tree, rather than in a forest where they tend to grow taller.
I’m a wee bit confused, that you moved in 3 years ago, viewed the house and garden presumably, saw this massive tree, bought the house, and now 3 years down the line you’ve decided the tree must go. The tree won’t have grown a whole lot in three years, so why have you now decided the tree needs to go?!
i would build fence that is not made of panels, so vertical slats. You can do it with gaps between them, or one of each side of the rail overlapping, or just up against each other (basically same as the gap method but you just put the slats right next to each other so the gaps are barely there. which means if the tree shifts and/or grows you can remove slats or cut the rail etc, rather than having to faff with cutting panels and that sort of thing. I would also say go pressure treated so you don’t have to paint it and it ends up a nice sort of wooden silvery sort of colour.
make the tree a feature, beech’s are beautiful trees (to be fair, as are most trees) especially a beautiful mature tree. Don’t fight against the tree. Accepting the tree and working round it, will be a much cheaper, less hassling and just an all round better way of doing things.
a tree that old will be an absolutely bitch and maybe 20 thousand other bitches to cut down, remove as much of the roots as you can, plus ridiculously expensive.