That specimen appears to have terrible structure. Looks like three dominant leaders with tight angles/included bark. Very close to the house for a specimen that, while a slow grower, will mature as a pretty massive tree.
If the client feels the tree "takes up too much sun" perhaps no tree in that location would be best. Quercus robur will develop into a dense, wide and shady canopy.
Removal and replacement with a better specimen(single dominant stem) farther from the house, or a smaller maturing tree might be in order. Otherwise, removing a smaller scaffold close to the house and subordination pruning for two of the stems would be in order.
You will have to go see the tree to tell how much was removed in last years thinning operation, but the canopy looks pretty dense to me. You could probably get away with another 10% thinning, after the clearance and subordination, focused on the house side of the tree.