Yeah, I certainly can't imagine I'd be the first with this problem.
Unfortunately, I probably won't be the last either. I found a 332T Broadcom-based card in one of our older spare servers and popped that in for a go. No problems at all. So it looks like for the sake of time we won't be solving this one, at least not right now. While we'd like to use the I350-based NICs, the 332T will work well enough.
Now I only wish I'd thought of trying the 332T a week ago....