Thanks guys. They've worn me down and I'm ready to say uncle.
They think (or at least claim to think) that I'm asking for the 35k bonus. I get the sense nobody is too familiar with the 10k anniversary bonus (which strikes me as a potentially problematic for those of you applying for the card when the anniversary rolls around).
It's possible they are being deliberately obtuse to get me to give up. But they keep referring to the sign up bonus and seem to be in auto-pilot thinking that's what I'm asking for.