It is uncommon that a large Oak tree would require pruning for arboricultural reasons - not uncommon that people convince themselves that a tree 'needs' pruning. Trees very rarely need pruning to achieve optimal structural and biomechanical balance - they have been doing it perfectly well for themselves for millennia.
It is the conflict of people and trees which creates the human belief that a tree needs human intervention - it doesn't.
I suspect, from what you describe, that what you want to do is reduce (what you believe to be) an overbearing or excessive crown spread? This will provide no benefit to the tree which has optimised to the prevailing conditions. Done sympathetically the adverse impact may be minimised but be in no doubt, this is not to the benefit of the tree it is to the benefit of the human. Not at all uncommon for a short term 'gain' to result in a mid - long term loss when mature tree pruning is executed by inadequately trained and informed practitioners - very easy to make a complete ass of it and make matters worse for both the tree and the human when the tree reacts to loss of canopy.
In the situation you describe there are several options depending upon the current situation.
Has the PC had the land ownership record adjusted Yes or No? If No, they don't own the tree, don't bother talking to them about tree pruning since they don't own the tree. Talk to the person that does own the tree or take a unilateral action if the land is not in registered ownership.
If PC does own the tree don't 'fear' a TPO being initiated. TPOs do not preclude works to trees with a TPO they simply require a properly considered justification to be submitted to the LPA in support of a planning application. If the proposals are sound, reasonable and necessary, the work schedule will be approved. If the works are unreasonable, unjustifiable or unsound they will be refused.
TPOs are meant to protect public visual amenity but that is only 1 of several fundamental considerations in the decision making process. Don't 'fear' a TPO, they are for the greater good and simply require sound arboricultural justification for sensible works. There are also exemptions where consent of the LPA is NOT required for works to a tree subject to a TPO and suitably qualified and experienced arborists will be able to inform you if these apply.