I can help you with this a bit!
The easiest way to do it is have your chickens or geese over saturated in the same spot not moving until they literally pick everything clean. This way can take a bit of time but is the least back breaking.
You could even purposely move a roving pen around these spots until literally there's only mud and dirt left. (But where you'll have to be careful is the spring where the newness of life brings in fresh sprouts.
Another trick is to lay down black plastic over a spot for a period of time until there's nothing there that's alive. (Allow some time.) Then put in fresh sod of something else.
When you do the fresh sod, you'll not want the animals on it etc.
Now what the cattle people can tell you is that they cut down the goldenrod in the fields because if it gets too thick it drowns out the grass from having sun. (I'm curious if someone did this naturally with just planting goldenrod could they use the golden rod to get rid of the bermuda grass...?) (But one issue with this is that golden rod is a fall plant. And also when these people talk about this they are talking about longer grass not 2 inch grass.
Another thing you can do that's mental rather than physical is look up what heights the bermuda grass doesn't do well when cut at. And look at whatever height the other grass that you do want in, can flourish at and see if there's a vector ideal height where the other grass will flourish but the bermuda grass won't? This is more of an experiment, but it might be fun to try.
You could try changing the lawn mowing height also. One of the reasons for this is that in the spring any grass height is going to work. But in the hot dry summer months, the shorter grass will have a harder time. Especially during the really hot dry months the non-bermuda grass will have a harder time than the other grass. So during these really hot dry months, you'll want to let the grass be cut at taller height rather than a shorter height because that helps the grass endure the summer heat better. Bermuda grass is pretty tough stuff also unfortunately. In the fall you can reduce the mowing height again. (But a side issue is that in the deep south a lot of insects like taller grass height also :S. But you can counter the bugs with ducks, etc.)
Personally oversaturating the living grass spots with geese, chickens or black plastic in an organized manner over time might be less labor than going over every single spot. And because you probably don't have that many chickens or black plastic etc, you may want to do this over time in sections. (But a problem with chickens is that they like to roam and will be seduced by 'the grass is greener on the other side of the fence' and try to move themselves.) Ducks won't be as tempted to move but they don't like grass like geese and chickens do.
If you want to just lay down old plywood sheets, you can watch free ads for when people toss these from redoing the roofs of their garage or house etc. And then you lay them down over spots where you want to kill the grass.
If you manually dig things up, its really hard when the soil is dry and hard. If you run the sprinkler in the early AM, and then let it sit a couple hours, and then dig it up the labor isn't as bad.
Another alternative is to lay down seed of another plant that will outgrow and overcome the bermuda grass, ...but that could be as bad or worse as the bermuda grass.
And having all the whole several acres be a dirt desert is also something you don't want to do also. That would also take out the water in the soil and make it hard to live. I'm thinking you could lay down some trees in that area also, that will help you make the whole area vibrant and beautiful. (Do aspen trees grow in your area? I'd lay down some of those if they will grow there, as they are great shade plants. Some apple and plum trees are good for people and animals. And if you want to do honey bees, ever, black locust acacia trees are amazing if they'll grow there.) I get that you want pasture space, and I'm not going against that, but grass doesn't grow from scratch well in terrible heat and out of season. And you want the whole area to flourish. Way back in olden times they'd space more variety of plants in a pasture, and not like the modern model where everything is ONLY one type of plant like alfalfa etc. They might put raspberry bushes or some other thing on the sides of the acreage etc, or other things. And there'd be wide variety of pollen types.
. Once they are taller than everything else use a wick applicator that will hit them and above the stuff you want to keep. I have one that I mount to my tractor bucket. I’ve eradicated Johnson grass with it.
