I’m a big fan of clover, and while I do not have experience with this particular species- Dwarf White Clover might be one to consider? I can’t speak to how thrifty it is with moisture, but it should grow low, handle sun and shade, and would also contribute to disease suppressive soils through its legume/redox properties. It should be vigorous enough to suppress weeds (it is considered an invasive in some ecosystems/areas). Clover is also great for beneficial insects, and I also can’t speak to its effects on tree health and immunity- but I would hope that its disease suppressive characteristics would be a positive in this category.