You don't say where in the south you are from, but in our part of the country (very bottom of zone 7B), red tips are constantly plagued with all kinds of fungus and mildew problems. I wouldn't recommend them to anyone here.

If the shrubs are large, I wouldn't advise you to mess with chemicals. I would get a good mattock and chop the crowns out of the plants, and then begin digging out the roots. You may be very pleasantly surprised that it's not as bad a job as you think. (You could also find someone with a winch and pull them out of the ground.) I just finished a project where I pulled out overgrown cleyera. The roots were large, but they weren't matted in the ground. I simply cut each one with loppers as I came to it and they came out pretty easily as I dug the bed. I had the same experience with overgrown hollies about ten years ago.

I just think that Round up may not be very effective on a large, overgrown shrub, and may very well take multiple applications. When you consider how easy it is to let it drift onto and kill plants you want to keep, I'd sharpen my shovel. Remember, Round-up is not effective unless the plant is actively growing.

These are my opinions. It will be interesting to see what others say.

What is your problem with using Round-up? If you are afraid it will get on other plants, then here is the solution. Cut the bushes as close to the ground as possible. Put a 3/4 inch paddle or spade bit in your electric drill and drill holes in the stumps about 1- 1 1/2 inches deep. Fill the holes with Round-Up BRUSH killer. Do not use weed killer, use brush killer. I use the concentrate. The plants will be dead in no time.

Much like leyland cypress (plagued by bagworms and drought canker), redtips were over planted and attracted a fatal disease. It will eventually get to your plants, I would shovel prune and move on to something better.