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.

Could be rye grass, which is used to oversees bermuda when it goes dormant.

That is bermudagrass and the only way to reduce its growth and development in your St. Augustine is by promoting an environment that is more beneficial for St. Augustine and less beneficial for bermudagrass. In addition, the use of Atrazine herbicide at specific times of the year can help reduce your bermuda encroachment. I'm not nearly as experienced with St. Augustine as are some others on this forum, so I'll step aside and let them guide you to St. Augustine success.

Hiring someone is out so it looks like we may be digging it up ourselves. I thought about chopping it low, then either trying to dig it by hand or round up it to death. Problem is I do have plants nearby I don't want any chance of being poisoned and I'd like to use the space for replanting so I've been hesitant on the round up. Do you have any idea how deep the roots might be? We just removed some stumps from three rose of sharons I chopped and round upped to death but we had access from the road so we hooked on to the stumps with my truck but this clump of grass is in the back so not so much access to it to pull it up.