Wait, I was wrong, Bitchy is not always the same card.
After more shuffling around, it turns out it's always a card in the first OR second PCI-E slot. It just happenend to follow that card the first time and lead me to label Bitchy as the culprit. I'm sorry I doubted you Bitchy.
I swapped 6pin cables and it didn't change, although it doesn't mean much on a single-rail PSU.

Past experience tells me it's the PSU. It's always the PSU. Because it's the only part I don't have a spare for and Murphy's Law is a bitch...

Still interested in your experience with similar problems though.