I feel bad for you, and I really do not have to much
in the way of suggestions.
I have mostly grass, so I am lucky.
In the dog run, I have pea gravel, and I hose it
off once a week because of the pee smell.
I wonder if their are some sturdy lawns that
grow in shade? What about moss? I think
that is clumps of grass that like shade and
water. Not sure if St. Augestine will grow in shade,
but if that does, that will withstand any pee.
Be carefull with store bought mulch, some of it
has chemicals that can harm animals.
Any groundcover that has flowers will
attrack bees.

How about checking out you local nursury and
asking them for suggestions. I bet they
will have some good ideas.