My suggestion would be to sow a strong growing perenial grass seed, such as farmers use to harvest a silage crop, because looking at he photograph it looks to me as not only is yout tree denying sunshine ( photosynthesis ) to the existing grass, it is also taking what natural moisture, therefore ---food --- there is availible, your watering the grass is similar to you drinking water, it quenches your thirst but doesn't feed you.
If you were to buy the "municipal" grass seed, that which you see on golf courses municipal gardens etc, I think you might have a problem establishing them because in general they are far more delicate than the robust perenial grasses, and you tree would be too greedy for them.
Grass feed is based on N.P.K. ( basically nitrogen -phosphates and potash) I would suggest you buy a grass feed with higher contents of P. and K. that will feed the rooting system and not make the grass shoot up . In the winter, if you have an open fire, spread the ash all over the lawn, don,t worry, the rain will wash it in. Feed the grass on a fairly regular basis, not too much at a time thoughand then water afterwards until it is established.
No matter what you do though you will not have a carpet of green next spring, lawns take time to establish.
If you go to your to your garden centre and can't find what you want then go online and buy, if that fails, ask your relations to bring som over next time they come. ----hope this helps you.