I may get shot down for this, but I think the issue is often that these skills are taught solely for the purpose of passing an exam. (I'm not saying this of you, Stringshed, as it's obvious that you have put a lot of time into it, it's more a general observation of what I've found with a lot of teachers.) And that attitude is passed from teacher to student - they see it as an unwelcome add-on to their "real" learning, as opposed to something directly relevant to them. My pupils are split fairly equally between Trinity and AB, but I still teach all of them the same things, and that includes working on listening skills, pulse and rhythm, and yes, singing. To me, a musician who can't sing is an oxymoron. Now, preferring not to sing in front of people and therefore preferring Trinity aural tests, that's absolutely fine. Also I think the AB tests can be too much of a memory test - singing back a long phrase is of limited value IMO. However, I think not including singing at all, as if it's an irrelevant skill for a musician, is really, really sad. I loved the question that used to be in the Trinity Initial aural tests, where they had to sing the tonic at the end of a phrase. No great feat of memory needed, and no large amount of singing, but just a simple and musical exercise. My ideal aural tests would probably not look much like the Trinity *or* AB ones! 
But they would definitely include singing.
I try to incorporate Kodaly in my lessons where I can, and if done from the beginning, even those who naturally find pitch matching tricky can become confident singers and sight-singers. It does mean making a choice to make it a priority in lessons - and in practice at home - and not just something done in the few weeks running up to an exam. In my ideal world, all students would have Kodaly lessons for a year or two *prior to* learning and instrument, and preferably in addition to once they've started. Of course, that's not feasible for most! But I do what I can. Even when circumstances have meant I do only have a few weeks to work with someone before an exam, I've had some real grunters make a pretty credible attempt at the echo tests after just a few weeks of working in solfa. But it's so much better if it's done from day one - then it never becomes something they "can't" do, because they always can.
If you've never look into the Kodaly method before, please do! I used to find it so hard to teach aural skills, because they came so naturally to me as a child (largely, I believe, because of the type of music education I was fortunate enough to have). I just did it, without (knowingly) having to work at it - so I had no idea how to help someone who struggled. Kodaly was revolutionary for me. Now I had a way to teach these things to absolutely anyone, and it *works*! I honestly can't recommend it enough. 
I have / do still sometimes have students who really struggle, and especially if they've not been with me from day one, or with adults, or those who refuse to sing at home (making progress very limited), etc - yes sometimes I've had students score poorly, and it's hard. I think that's the same with students who don't do so well in any part of their exam - we're bound to feel bad about it, even when both we and our student have done their best. I would focus on the positive with them.
Even students who are fine with singing can do poorly in the aural tests due to not having a great memory, or not doing well under pressure. It's tough. Tbh, in the case you mentioned I would potentially do both - keep working on aural skills, because they're so important, but work on them for their own sake, not for the sake of passing an aural test. Start at the level she is comfortable with, and work up from there, at her own pace. And then when the next exam comes around, give your student a choice of exam boards and see what she prefers. If it takes the stress out of it not to have to sing in the exam, then it's worth it to switch boards. But do encourage her to keep singing anyway! 
Oh, and in response to the later posts, fwiw I've found my students find the clapping the pulse far easier in the AB tests than the Trinity ones, because it's much easier when there is an accompaniment than when there's only a melody. Naming the metre is another matter (especially telling the difference between 2 and 4 time, which sometimes I'm not even sure is possible). I do also find that playing the piece more quickly helps beginners to find the pulse rather than the rhythm, and avoids them clapping quavers instead of crotchet beats, etc. It's possible the examiner was trying to play more quickly for that reason? There's helpfully fast and unhelpfully fast though!!
but it just doesn't happen. 
