Well if you own the land maybe you can use PD rights to put up boundary treatments- whatever the land is classed as...
If its a relatively new estate, it might not have any PD rights. However, there would be no fee to apply for permission on something that would ordinarily have been permitted under PD, if the rights had been revoked as part of the planning permission.
I think the OP should press for details of the relevant condition or landscaping plan that identified this area of grass as amenity space. It's not an unreasonable request, and the plans for estates can be complex and involve multiple submissions, so online services can be very difficult to navigate to find a nugget of info such as thing. As professionals who have dealt with the application, the planners would be better enabled to find the relevant detail; something they should have been doing rather than cutting and pasting together a document of irrelevant vandalised alleyways being incorporated into gardens.
I don't know whether this would make for suitable material for a FOI request as technically the info should already be freely available, but there's no fee for asking.
There may be no such documented evidence, and thus the council may be casti the assertion based on what the "spirit of the design was supposed to communicate" about a feeling of openness and sense of place (blah blah), concepts which may have relevant lip service in the local planning policy. As such, it may entail having an adjudicated argument as to whether putting up a fence between your house corner and the neighbours garage corner makes the estate as a whole look more like an overbearing wall of red bricks. Try to see this from an inspector's viewpoint, though it's a relevant argument that your enclosed private space is much smaller than your neighbours due to the triangular nature of the garden. There is a minimum amount of private amenity space specified in most planning policies to ensure developers don't overcrowd land with houses by making postage stamp sized gardens. If your area is less than the minimum, and less than that enjoyed by neighbours, it's reasonable to argue for an increase. You're specifically disadvantaged by it being split into two small triangles by the nature of the construction
On the plan posted there is a large swathe of public open space nearby. The council officer really wouldn't need to bring his dog to **** on your front lawn, though it does seem the owner may have inherited some of the pet's attitudes
Last point; how well do you get on with the neighbours? Any scope for taking the fence down between the rear gardens and then both families enjoy a bigger area? Sure, there are plenty of things that could ruin this (pets, particularly untidy or boisterous football playing children..) but it works from time to time. I share a back garden and garage with my neighbour..
