"Early morning watering, completed about 5 AM, is recommended."   ~ Q

This is more of a cultural consideration that a horticultural one. The peak demand for water is probably from 5:00-9:00 in both the AM and PM.

If you want to avoid that time period, try 9:00-10:00 in the morning. Lawns are often a serious investment and deserve to be treated well. They can raise the value of your property and be a source of pride for the whole neighborhood.