Yeah, a HD 6570 or a HD 5670 is going to be okay, but there isn't any doubt that the bottleneck would still exist.
The PCIe slot is likely to a e a 1.0, meaning less bandwidth.
Also, what are the specs or your PSU?
The power rating and the amps on the +12V rails.
You can find that out on the side sticker of the PSU, mentioning about the power specifications.