My dogs like to dig around in the pinestraw for twigs and stuff to chew on. If I were to use bark or chip mulch they would go to town with that and eat it all. A possible solution to try is some alternative mulch materials like shredded roof shingles and rubber (from old tires). They use that stuff in playgrounds a lot.

Maybe some bricks or tiles or maybe some RR ties embedded in the soil where you could make a little patio sitting area.

It might be juglone from the tree which I believe pecans have but this has been debated before. What type of grass have you attempted to plant?

Have you considered mondo or monkey grass? It only needs to be mowed once a year and would hold up to dog traffic. It likes partial shade.

I have a place like that in my front yard that I planted with liriope. Dreadful soil packed so hard that even weeds didn't grow. It takes about two years for liriope spicata to fill in and cover it completely, but it grows where nothing else will.
On the other hand, I have a shaded out area in the back yard where even liriope won't grow - so I would give it a try in a small area first to see how it does.

I have the same area as you and have been trying forever to figure out what to do with it. Eventually I'd like to brick it or build a deck over it or something. This spring I started planting mazus reptans around the border of the area and it's starting to fill in nicely. It seems to hold up to foot traffic fairly well. Only thing is, it has to be watered about every other day...And it does not like being peed on...Good luck and I'd love to hear what you end up doing!

If you decide to plant something, till it up as deep as possible to give whatever you eventually plant a good start. In addition to monkey grass, ivy might work well also. Try them both and see which one wins. My dog love sniffing around in the monkey grass and lying in it.