I actually don't criticise Allegra Stratton for her levity in that recording.
If you are developing "lines to take", or practicing your communication skills, it's an intense process, particularly when doing it in a proper studio environment, you try out multiple different ways to address tricky questions, you watch the recording of yourself cock it up in five different ways, and the need to break the tension is quite legitimate, particularly when you know that the thing you are developing lines to defend is actually quite hard to defend. I seem to recall that President Bartlett did it more than once (we know that because Aaron Sorkin documented it for us), and I'll bet most people in that situation have done it. (To be fair, you would rather hope that you could trust your colleagues not to leak it).
So I don't particularly blame her for either what she said or for the style in which she said it. No, I blame her for choosing to work for an outfit whose objectives and culture led to her being in this situation, and that's why her tear-streaked resignation elicits little sympathy from me. If she hadn't knowingly chosen to work for a mob who think the law doesn't apply to them, she wouldn't have been in this position.