It does help, though, to remove much of the brambles' top growth before attacking the roots. ...
Yes, it's a good idea to cut back the very long stems of the bramble first, to allow you to get near to the root. But it's best to leave a couple of feet of stem, to give you something to lift the bramble out with. It really is quite easy. This is the tool:
55 sq yards is not too large an area. To get rid of the grass, you could simply dig it over, removing any large tufts. (You can use a mattock to remove tufts of coarse grass). Then you could perhaps plant the area with comfrey or potatoes, some other robust ground cover, which could be dug in at the end of the first season.
And make sure that you use very strong gloves when handling brambles.