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I suggest you contact your local extension office (found under gov't in your phone book) since rust is usually caused by a fungus, they will be able to help you find out exactly what you have and be able suggest the best options to control the problem. This article will give some background info:

https://www.gardeningknowhow.com/ornamental/flowers/roses/yellow-rose-leaves.htm

This is generally a sign that the new sod is not making good contact with the soil below in spots. You may need to roll the sod again so that it makes a good contact. I would also increase watering to the lawn, to help encourage root growth.

Your best bet would be to dig as much of the fungus out as possible and treat the area with a fungicide. Repeat with the fungicide treatment once a week or two for a few months after this. Yes, placing the organic lawn soil to the area probably helped them grow back as they need organic material to grow in.

If you have not read it, this article will also help:
https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

I would treat the lawn with a fungicide. That should kill them. This article will have other suggestions as well:
https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

I would overwater that area to wash out the excess nitrogen, which is what causes urine spots to go brown. This will allow the grass to regrow.

This is normally an indicator that the lawn has gone dormant, which indicates that it may not be getting enough water. Consider increasing the water.

If you think the lawn may be getting enough water, you may be over fertilizing, which is burning the lawn. Back off some on the fertilizer if you feel the lawn is watered well enough.