Wouldn't you know it, it works fine with SDK 2.1
I used to have the same problem, and SDK 2.1 does indeed fix it
The driver version is 10.12 as it is the latest that works on Gentoo, with Linux kernel 18.104.22.168.
I guess I was reluctant to try this SDK, since 2.2 worked much better with HD5570 (also using an older version of poclbm), but a few Mhash/s is not a bad price for getting the full setup to work.
As a nice bonus, CPU load is way down. With SDK 2.2, each miner consumed 100% CPU, but now it is barely noticeable.