I was thinking the same thing. You can't get rid of bermudagrass without killing the St. Augustine. The good news is the St. Augustine stolons grow a half inch a day in the summer. So killing the yard and starting over wouldn't take long. You say the front yard is nearly perfect so in the perfect areas use some of your yard as a nursery and plug the back. The front will recover quickly and the back will soon be looking as good as the front. The key is using the top labeled amount which could range from 3 to 5 ounces per gallon. Then to use it as soon as you can then in about 2 weeks. You should have pretty much wiped out the back lawn. I would then cut the grass as low as possible then plug all you can. Keep it well moist and then wait. I would think by the end of next year you should have a solid back yard. Good luck.

Get a non corrosive pale, mix together some stump and brush killer with diesel fuel. Take care to wear protective clothing so none of the mixture gets on your skin or in your eyes and nasal passages. Use a paint brush to apply the mixture to all the foliage and also any bark of the roots or sucker trunks. Keep brushing the growth you want to kill untill your mixture is gone. Do not leave any mixture in the pale that will have to be stored, and do not pour any of the mixture onto the ground. Only use like I mentioned above, by applying it by paintbrush onto the trunks, bark, and folliage of the trees and shurbs or weed brush that you want to get rid of. After finishing the first major application; safely discard the pail and brush, along with any empty containers, and clothing worn. Shower well and apply Aloe Vera, followed by a good moisturizer to your skin. I know it is a little late to tell you this, now, but your job would have been much easier if you had drilled holes in the trunk after cutting down the tree and applied this stuff; waited about a week, drilled some more holes and applied this mixture again. At the same time any root suckers should have had this stuff applied. Only after doing that long enough that you could no longer see any sprouting from the trunk and suckers coming up from the roots should you have had the trunk and Major roots ground. Even then you may have still needed to apply the mixture to some sprouts, but it would not have been as many as what you are seeing now. Also keep in mind that every time you cut off one of those root suckers it sends a message to the root to produce even more new suckers. My experience with the mixture I described above, and when it was applied only with a paintbrush, never damaged the nearby ponds' ecosystems or polluted the soil. With the method I just describe, I successfully got rid of some horrible hackberry trees, which grew in a thicket form, on our side of a farmer's fence row. That thicket was an about 5 foot deep strip which ran all along the fence line of my property. At the very end of that thicket grew a large infested chinese elm which dropped sticky sap and also sent up many root suckers, which if allowed could have also created a smaller thicket. First we had all the trees cut down and removed. Then I killed the stumps in a manner described above. then after a couple weeks I had the stumps and major roots all along that fence row ground. All this was done in the Summer, and by the fall into the following spring, no more suckers grew for me have to contend with. That following Spring I successfull planted a line of more desireable oaks, pecans, Foster holly, and Loblolly pines. All the those trees have survived, except for some of the holly which ground squirrels killed where the Elm tree's roots had been ground. My only other loss was two loblolly pines in front of there where the water table rose too high and suffocated their roots. Last fall I added two Ponderosa pines back in the area where the old elm had been removed, and they have done great ever since. I first had some sandy loam top soil brought in to help raise the soil more above the high water table in that area. Also, chose to not replace the two loblolly trees which had suffocated from the water table that previously had been too high.

Very easily. Inject them with 24D, the active ingredient in Ortho Weedbegone and other weed killers. I use a small syringe (1 cc). Inject about one cc of Weedbegone into the center small pups. Or in larger plants inject more. I drilled a small hole at a downward angle in larger ones and infected several cc s. Large plants will die at the root in a week or two. Smaller ones will turn brown in a few days. In inject an inch or two from the base. It is easy to control groups by working from the outside in, killing what you want and leaving the remainder. I just finished thinning mine out.

I have never had to take a banana tree out BUT I did remove 2 oak trees this spring by using a cordless reciprocating sawsall with a brush blade from diablo and I cut into the dirt and used a sharpened hole shooter shovel ( a skinny shovel) to lift them. I bet the saw will separate between the trees pretty well. The saw cut right through the dirt. I used a cordless but your power source is not far so one could use a corded sawmill. I would put a tarp down on that pretty rock work to keep the sap off of it. Duck tape it downs it won't move. Sawsalls are GREAT gardening tools