I've had similar problems with sunny areas and tried watering more. It helped a little. Then one year I tried fertilizing (manure + miracle grow) and my lawn turned completely green, even the full sun areas.
What's the specific importance of a green patch outside your home?
Even in Ontario, not particularly hot or dry, people water to prevent their lawns from turning brown in August.
Using water during a water crisis so you can have a green patch next to your driveway is the most excessive American thing I've heard today (a worthy winner in such a competitive field)
While the concept of using perfectly potable water to water a lawn is a little bass-ackwards, people do do it here. I personally don't, my lawn goes dormant in the heat of summer and that's ok too.
My 'patch' of grass here in Texas is thousands of square feet. The lot my house site on is .33 of an acre, which in metric is 1335 square meters. The house itself is ~3200 sq. feet (306 sq. meters) over 2 stories so the foundation is roughly 1/2 that so you can extrapolate that is a crap-ton of area left for our patch of grass.
This is in no way unusual or extravagant for the suburb I live in and provides lots of enjoyment for our 2 dogs and 6 kids.
Our Texas adapted grass goes dormant half of the year as well so the actual consumption is relatively minimal.
Our water use is fractional of a traditional manicured English garden.