Everything I've read says to water deeply over a longer time than short bursts. That builds resilience and root depth. Watering during the day is no bueno because more than half of the water evaporates before it gets a chance to soak in deeply.
I'm not trying to tell you what to do but you're trying to keep alive something that wants to die while wasting a ton of water and probably causing you a bit of stress.
I think you might find some benefit with top dressing your lawn and aerating (if you haven't tried that already) to build the soil.
Aeration gets water and oxygen to the roots and breaks up the ground.
Top dressing adds organic matter to your soil. If youre like most of us and don't have good soil to grow in you're fighting with one arm behind your back.
Per every 1% of organic matter you develop in your soil, your water holding capacity goes up.
So say you have an acre of land and with top dressing your able to add just 1 % of organic matter in a rain event that you get 1 inch of rain your soil gains the ability to hold thousands and thousand gallons of water.
I've seen studies that show 16k gallons of water per inch per acre and some that say only few thousand. Some people claim up to 20k gallons. Not sure what's right but the point is improving your soil is paramount.
Here's a MSU study
https://www.canr.msu.edu/news/compost_increases_the_water_holding_capacity_of_droughty_soils#:~:text=Water holding capacity of soil,down to one foot deep.
Good luck