A. Boy you do really have a stand of bermudagrass. Because it is so dense and so much of it you might want to consider managing the Bermuda instead of trying to get rid of it. Bermudagrass has a very high requirement for sunlight, more than other grasses. You might want to consider mowing it, weed whacking it or burning it instead and manage it as a stand of grass that will never be erradicated. Of course it will never be a good solid stand of grass without irrigation and I am not suggesting that.
Organic controls include burning it down with organic chemicals. Vinegars with high acetic acid levels will help burn it back. Of course it will come right back since the roots and rhizomes, stolons are not controlled.
I have not had much luck with the standard vinegars since the acetic acid level is too low, around 5% or so. They do not cause much plant damage when I have used vinegar made from coconut or palm even at some higher values. Vinegars with higher acetic acid content are available from retailers on the internet but be careful using these products. At this level of acidity they are very caustic. You must use all plastic or stainless steel sprayers or they will corrode quickly. Vinegar is expensive.
One method I have used at a small farm out of state is fire. I have used a propane and fire torch
with great success. This is a great price since I paid about $119 for this same torch last year.
You must be VERY careful with this since you are hauling around a propane tank, rubber hose that is pressurized and a fire torch. You get the picture. I would burn it down. Be sure to mow it first or cut it back to reduce the potential for fire. It does not cause much smoke so it should not cause much alarm in the neighborhood. If you find that it is a little too flammable then wet the grass with a hose before you torch it. This will cause more smoke but the fire is more controllable. With a potential of 400,000 BTU’s it will kill wet grass. Torches of lesser capacity may not. Have someone on guard with a hose in case it does catch fire but it most likely will not.
The next thing I would do is find some old carpet. Natural fiber backing is best but synthetic backings will work. Cut it in squares about one yard square. Cut a slit to the center and cut a hole out of the center for the trunk. Lay the carpet on top of the grass and it will shade it and kill it. You can use the torch right up to the edge or the carpet. You can cover the carpet with wood mulch and it will look better.
Carpet allows water and air to get to the soil and roots of the trees. Putting plastic down does not. I have a problem with newspaper as well. I tried newspaper and it didn't work well for me. The carpet basically smothers the grass while still letting the soil "breathe". It will work better than a weed barrier but it will not last as long. There are some chemicals in the carpet that will be released as it degrades. That is the tradeoff.
You can also use about four inches of wood mulch over the grass. I would mow it down as low as you can before spring and lay the mulch on top. The grass will come through the mulch and then you must spray the grass as soon as you see it so it does not get to the light. Once it gets a few inches exposed to light you wont start starving out the roots. The idea is to provide shade with the mulch, eliminate new growth as you see it so it cannot emerge and rebuild itself on top of the mulch.
I hope this helps. But I like fire.