I know how you feel.
I have a mushroom problem in the winter and spring.
(and I only have 1/3 acre)
Every morning I do a lawn check to pull up any mushrooms.
Their is a fungus product I have seen at the garden shops, but
I never checked it out.
I am afraid to put anything on my back lawn since my dogs eat grass.

Your neighborhood nursery might be able
to tell you if their is a animal safe product on the
market.
IF their is, even then I would jot down the manufacturer and call them and double check.

Let us know what you find out.