Farhan Ahmed wrote:why not make a flower bed here?
Add DP/phosphate.
Bermuda grass should do well in shade.
Water regularly
plant some more patches of grass.
That seems to be an appropriate plan of action for best rectifying the current situation however if this fails or is too much hard work im afraid I would be very tempted to start again from scratch and create something modern, contemporary and eyecatching.
Seems the nutrient/moisture sapping trees and your poor soil are working in unison to inhibit the growth of a lush green lawn, why not think totally out of the box and if you can afford the time and resources build raised beds so that any plantings are up and away from the tree roots with good nutritious soil. Don't be palmed off with subsoil as it may look good but its guaranteed to be infertile, taking time effort and nutrients to make it fertile again.
As you already have good structure in your garden with established trees and the architectural elements of your boundary wall. I would emphasise their qualities and maybe paint them in colours that make the plants you have pop out at you, green foliage or blue shade flowers look amazing against a backdrop of terracotta walls.
Alternatively mid blue walls with nothing but white theme flowers will make the garden appear more spacious, calm and restful. The second advantage with such a colour scheme is that during moonlit nights the white blooms against a relatively dark background give off a unique shining luminosity that almost rivals the glow from a full chandni moon. It really does look amazing but you would have to be extremely ruthless avoiding the use of other colours for that daytime dreamy look.
Some ideas for you to consider in respect of coloured walls and raised beds