I only see a problem if you want to run either 2560 or an Eyefinity setup. Other than that, the old but modified Tomb Raider engine should be no problem for your machine to handle.
It can depend a little on how much they have decided to implement DX11. 5770 is a pretty weak DX11 card and since it was recently announced that they are working on DX11 implementations, this could be a problem. However this late in development I dont think we are talking massive amounts of tessellation, rather something else, maybe some shader and effects implementations.
Did you do the hacking mini game at that website? from what I've seen, the minigame on the site is exactly like the one in the game. Could be interesting if you're trying to hack something in a crunch..