I have the blackberries to thank for the eradication of scotch broom from my back yard. They do indeed choke everything else out. Luckily I have nearly 7 acres to stash the corpses on, and I long ago decided to manage my cleared acreage as a meadow rather than as a lawn, so piles of berry canes returning to nature are not a problem. If I were in the situation you describe, I would rent a shredder/chipper and reduce them to mulch, pack the mulch in a plastic garbage cans and haul them to the dump.

I think your suggestion will be my ultimate solution. I had planned to buy a chipper/shredder/mulcher, but found that a decent one that'll handle both big and small branches costs ~$1,200. That's a lot of sample libraries. Which is how I price-compare nowadays: judging any expense by its equivalent in musical instruments or software.
 
Unfortunately, renting one isn't cheap, either. About two SM-58's per day. 
 
 

Memorial Day Weekend - I collected as many blackberrys as possible and made a cobbler. X said it was amazing and ate most of it.    I bought vanilla ice cream to go with it so I had what I needed. good times 

Ortho Ground Clear.
That's what I usually recommend to customers with unruly brush problems.
Better living through chemistry.

Seriously? What's that going to do to the lilacs, camelias, holly and cherry trees that co-habit the space?

They will accompany the blackberry bushes to the great garden in the sky.
Sorry, Bit, I didn't know there were friendly plants there.

Have tactical nukes been ruled out yet?