Southwest Missouri? We moved to SW Missouri from Oklahoma back in the late 1940's, a little town called Pierce City. We did the same thing; all of us went out and picked blackberries and were eaten alive by the chiggers. They are vicious there, especially if you have no immunity. The locals told us about the sweat sock/kerosene trick, and it seemed to work. Still picked up a few bites and a tick or two, but we were able to pick blackberries and survive.
I loved it there, but the winters were horrendous. You could almost live off the land; blackberries, boysenberries, Hazelnuts, walnuts, hickory nuts, and even strawberries grew wild. Lots of game; rabbits, quail, squirrels, lots of good fishing as I recall; there were orchards where you could pick your own peaches very cheaply and you could grow most everything in the gardens there. Mom canned a lot of stuff, so we lived on the cheap.
Have you thought about mowing and spraying the ground around the blackberries? If they are accessible, that might be an option. They do seem to grow best in hard to get to places, though, like in the fences along the right of way and on the dams of earthen ponds. Let us know how you make out.
P.S. You might ask some of the old timers there; I'm sure they solved this problem years ago.