So, a further DP. Ms SWMBO Stanbury just attempted web check in for her flight to CDG tomorrow, which had 2 TK and 2 HK status flights. I advised her that this would almost certainly be a fruitless exercise, and we would have to either call AC or wait until she is at YWG tomorrow.

Result: the check-in worked like a charm! She now has BP's both printed and in her iPhone wallet - the latter obtained through the app.

Cow's tool still shows the 2 TK's.

I don't begin to understand this, but "ours not to reason why" I guess.