OK, no more fashion talk, I promise.
SFO, I want to go back to something you mentioned earlier...that you are looking for a neighborhood that will be comfortable for your dog. I don't know how large your dog is and how far it is willing to walk for its relief breaks. (My dog has short legs and is a dawdler as well, so our walking radius from home is usually pretty small.) In central London you may have to walk some distance before you find a patch of dirt or grass.
On your map of London you will have noticed a lot of green spaces, many of which are public parks, but you should keep in mind that some of them (and in some neighborhoods, for example South Kensington, most of them) are private, for residents of that particular square. And many of the private ones do not allow dogs, so even if you live in that square, you won't be able to bring your dog into the shared garden. Or there may be a waiting list. So you'll want to keep this in mind when checking out neighborhoods and flats. I did not find our agents to be particularly knowledgeable about the local rules, so I had to walk around and read signs myself, and a bit of internet research helped as well. For example, we looked at one place on a dog-friendly garden square, but when I emailed the local residents' association a very nice woman informed me that unfortunately the waiting list was rather long and it might be a year or longer before our dog could enter the garden.
Something that impressed me about Mayfair is that in addition to the obvious options of Hyde Park and Green Park, you are also allowed to take your dog into Grosvenor Square, Berkeley Square and Mount Street Gardens, which are all public spaces, and indeed I saw happy dogs being walked in all of these places.
You should also bear in mind that public parks, including all of those I just mentioned, are closed between dusk and dawn. There is a gate, and they lock it. So in the winter when daylight hours are limited, you will need to have a backup plan, like curbing your dog (using the gutter) or training it to use a litter box indoors. Or try to find a patch of dirt under a tree near your building, if you can.
I am not sure what the dog-friendly spots are in Marylebone, but perhaps someone else here will be able to advise you on that.
Happy hunting!