This may answer your question OP. It looks like you cannot climb the tree w/o permission so it will probably have to be the crane option for you.
Frequently Asked Questions - practical issues
Branches
Can I cut off overhanging branches?
Yes, provided it is done without trespassing onto the other person’s property. It is also permissible to climb into the tree to undertake the work, again so long as it does not require going into the neighbour’s garden/land. Note that trees covered by a Tree Preservation Order (TPO) or in a Conservation Area will require prior consent from the local authority.
Do I have to get permission from my neighbour or give them notice to cut off the overhanging branches?
No. Your actions are classed as ‘abating a nuisance’ which does not require permission. Only in situations where you need access to their land to undertake the work would permission be required. Similarly prior consent from the local authority is required for trees with a TPO or in a Conservation Area.
What do I do with the prunings?
Once branches are cut off they should be offered back to the tree owner. If the owner doesn’t want them then you will be responsible for disposing of the prunings; you can’t simply throw them over the boundary into your neighbour’s garden!
Can I cut back further than the boundary to prevent regrowth causing a problem?
No.
What if my neighbour complains about how the tree looks after I have cut off the branches to the boundary?
They do not have any legal recourse but in the interests of good neighbourly relations you might consider options for compromise, such as sharing the cost of a tree surgeon to create a balanced canopy.
Am I liable if I cause damage to a neighbour’s tree as a result?
Yes. In law you would be considered negligent. Sometimes branch removal can lead to tree failure due to disease, a change in the balance of the tree, or different wind loading that causes the tree to blow over. For these reasons it is important to employ a competent tree surgeon or arboriculturist who could minimise risk and would take on the liability for the work (check they have public liability insurance prior to engagement of services).
https://www.rhs.org.uk/plants/types/trees/the-law?awc=2273_1696782669_5f71cf4f175afb4485dd48a1a1a39aec