Urea fertilizer. Urea is used in in animal feed.

Aside from commercial fertilizers which I'm sure cannot do damage to a dog's paws and no dog that I know of will eat the granules, then that suggests "organic".

And one of the best organic fertilizers that feed soil and invites worms, and invites moisture to stick around, is

COMPOST.

Compost can be bought from soil dealers---but watch out.
Soil is generally an unknown product by the average home gardener. You have to depend on the dealer selling you what you hope the soil will do for you.
So, ..find a dealer that your neighbor or friend, or your local BBB has information on. Depending on the amount of compost needed to feed an area of lawn, get estimates on the amounts required.
Generally, one inch, laid over the grass will feed it for an extended time---depending on the quality of the product.

I might add....if you don't have a compost pile yourself---start one--it can pay dividends in the future.

You can estimate yourself how much compost to buy by simply measuring, length X width ---in feet....multiplied by depth .... in inches. That figure is multiplied by 3...and divided by 1000. (move the decimal three places to the left)
Example: lawn that is 30 ft wide and 25 ft long = 750 sq ft
Muiltiply by 1 = 750 multiply by 3 = 2250 ---divide by
1000 equals 2 and a quarter cubic yards.

Generally, any amount of soil more than 3 or more cubic yards should be bought in bulk and delivered by truck onto your driveway where you then move it by wheelbarrow to where you wish to spread it.

One inch spread over your lawn will not --will not---smother the grass. Grass will grow up and pull the goodness down where it will increase organic matter and invite bacteria which will feed your lawn and invites worms to come and stay. As organic matter is given to soil it will encourage moisture to stay and improves drainage.

Do this 1" every spring....and maybe--depending on your pocketbook--every fall-- you will help your lawn become the best it can be.