I think physical eradication will have to be your first order of business. I know that is dismal news, but unless you are willing to use an herbicide, physical controls are the most plausible way to be burr free. Or at least burr-less.
Maybe a combination of methods would make it less painful. Weedkiller + smothering + physical removal.
The herbicide would be the first prong and be delivered when the burr plants begin growing to be optimal. Timing is important because many plants will continue to produce seed if they have begun that process, even if a weed killer is applied and the parent plant dies. (Dandelions are one for sure that this happens to.)
If you opt for the herbicide route, don't use glyphosate, it won't give you the best results. use a product containing triclopyr, such as a blackberry herbicide.
That would at least give you a leg up on that years burr producers. Then burying under several inches of mulch. Wood chips work for me, but I do not have evil burrs. Then when they begin to emerge through the mulch again, pull them, or as I prefer, scuffle hoe them.
I have a tool that I love for weeding. I have had good success with perennial weeds by using this.
roguehoe.com
A very high quality well made tool that will last you for many years.
Yes, it just slices the top of the plants off, leaving the root, but it's fast and you can do it standing up. After several repetitions the plants roots will die if they can't store energy from the tops.
As to unwanted Bermuda... all I can say is I moved 1500 miles to get away from it.