Another update.
Changing the whitening algorithm to use xorshift64 yields results that look okay in diehard. Here's the details:
http://1474orchard.ca/projects/?x=entry:entry121002-012642
The xorshift I'm using is left 13, right 7, left 17. This is also a good PRNG, so it likely makes sense to have the code around anyway.
If anyone's interested in implementing this or studying it further, just follow up here or by PM or what-have-you.