You have to send a picture of your lawn, exactly where you live...need to know whether your lawn is warm season grasses or cool season grasses. Huge difference. I am guessing you have cool season grasses which will go dormant in the winter and with proper training, fertilizing and mowing practices should easily resist droughts.
How much illegal is it to water your landscape? If you trained your cool season grasses properly you would only need 1" of water applied to your lawn. Allowing grasses to 'die' or go 'dormant' to the point of live crowns and dead grass might work ONCE. One dormancy other than winter, which is never brown btw., is all a grass crop can handle without succumbing to weeds, compaction, hydrophobic soil. All that work to make a beautiful lawn will be wasted.
What do you mean there is not much soil? How much water is legal? What are the parameters used to call a critical no water your landscape order? There are many reasons why this order is simply wrong. Remember the dust bowls in central America? Watering of cool season grasses when done correctly can save water and still allow green grass that is using up CO2 so our planet doesn't become a Venus...I am being very facetious. That is not a problem.
When this stupid order is called off by the city 'experts' on water and landscaping in cities...grrrr...we can teach you how to train your grass crop called a lawn to be drought resistant to survive the next ban. Your lawn if cool season grasses can be trained to be luscious and vigorous on 1 inch of water per week. If you train your grass by watering properly.
Grasses (cool season) have basic maintenance practices every lawn owner should be aware and using. Warm season grasses are an entirely different deal. Cool season grass lawns are far easier to cultivate and manage to never ever need any other chemistry applied other than fertilizer. up to 4X per year. Did you fertilize before the ban? Do NOT FERTILIZE now. At all.
Your lawn should be mowed once per week. Mowing will not hurt or help your grass during drought. I am hoping your grass height is kept at 3 inches MINIMUM. 3 1/2 is best. If your grass is shorter than 3" do not mow. Aeration by pulling plugs of soil out of your lawn bed leaving them where they fall is a good idea to do now. If you have warm season grasses this answer is irrelevant.