Any possible solution to any problem has its drawbacks.
Your list seems to put a fair bit of emphasis on how much work different options entail. The least amount of human labor is probably the heavy machinery option, i.e., have it brush hogged down. Not sure whether your topography is compatible with doing that, but to the extent it is, get it cut down by machine, then pile it up by machine in a location where you can leave it for some time to compost.
Expect that you may need to cut back some new growth out of the pile set aside to compost, but remember that will be much less work than dealing with the original condition
Add other vegetable matter to the pile as available, burying the canes as much as possible over time and encouraging the decomposition process.
In the original areas, where the canes were cut back with the brush hog, expect some regrowth. Cut it back when it appears. Be patient and persistent.
The other relatively low human effort option involves using goats and that means putting in fencing. If fencing is an option and if fencing is something you see yourself wanting at some point in the future, then you might want to go ahead and put in fencing now, then use goats to manage the blackberry.
Personally, I would not burn it, nor would I have it hauled away. Why would I not keep all that nutrition for my soil on my property?