Ask Mr. Smarty Plants

The Lady Bird Johnson Wildflower Center, home of Mr. Smarty Plants, is dedicated to the growth, protection and propagation of plants native not only to North America but to the area in which those plants grownatively.

Since both St. Augustine and bermudagrass are non-native, we would recommend neither. Bermudagrass is one of the most invasive weeds of the South, and St. Augustine is a high-maintenance water guzzler. Not only that, but we wish you had not planted the Pistacia chinense (chinese pistache). Please read this Dave's Garden forum on the tree, particularly the 7 negative comments. Also, from Invasives.org, another article on the tree.

Now that we have criticized every plant you have mentioned (sorry, that's our job), we will try to answer the specific questions. If you have the St. Augustine because the lawn is in shade, there is not much native that we can suggest. Most native grasses require at least 5 to 6 hours of sun a day. Here is a previous Mr. Smarty Plants answer on solarization to get rid of bermudagrass. There are a couple problems with that: one is that it needs to be done in the heat of the summer and second is that it would be necessary to stop watering the St. Augustine if that is what you want to keep.

Here is another previous Mr. Smarty Plants question on replacing lawns with better, native choices.

Now, about the herbicides. The grasses are monocots, or narrow-leaf plants, and the tree is a dicot, or broad-leaf plant. You can buy monocot-specific herbicides that will kill the grasses and, theoretically, not harm broad-leaf plants. Theoretically. You can buy a dicot-specific herbicide to kill the broadleaf plants that will theoretically, ignore the grasses.  Theoretically. Or there are broad-spectrum herbicides that will melt the concrete. We don't like to use any herbicides because it's going to kill  something, it floats in the air,  goes where you don't want it and it's polluting.

If you are determined to resod St. Augustine, pulling out the bermudagrass and other weeds (and pulling them out, and pulling them out) is probably your only safe procedure. And when all the pistache seeds start sprouting in your yard and your neigbors's yards, and the football stadium, don't say we didn't warn you.