How fit are you? I got a plot a few weeks after the brambles had been burnt off it (they collect a lot of trash underneath, and burn green anyway, so if burning is an option go for it!), and cleared the roots myself. Established brambles tend to form a shallow stub root which is easier than you think to get out with a pick or mattock, just keep working the tool underneath and lever up from different points around the edge. It's very, very satisfying grubbing the ********* out! You'll also break the soil up a lot in the process so it's easy to dig afterwards (well, mine was). They make a cracking bonfire too!
I had no problem with couch under black plastic, except at the edges where it could creep in from the paths, but it won't kill bindweed very fast - you tend to find huge piles of ice-white bindweed stems when you pull the stuff back. I bet they slurp Roundup up when they are like that, if you feel inclined to use it.
Adrian