Season 5 spoilers, since I'm not sure about the chronology of this scene.
I'm not 100% sure if Claire has already been elected as VP, or is still in the running to become the next VP. Regardless, the answer remains the same:
Claire is projecting the image of someone fit to lead the US. Crying, while a sign of empathy and humanity, is also a sign of weakness. It implies that Claire does not have the stomach to consider a man's death without breaking down emotionally.
If she's that emotional, how can she be trusted to make decisions that could impact the lives of thousands if not millions of people. Will she be swayed by her emotions? Can she keep her cool and make the right decision, which is not necessarily the emotionally easiest option?
Also, you have to consider that in the HoC universe, no woman has held the office of President or Vice President yet. Claire would be the first.
It's very common for people to revert to gender stereotyping (in this case, Claire being overemotional at the funeral) when the voters are looking at Claire, the potentially first female VP, and considering whether they should vote for a woman or not.
So the answer is that Claire publically crying could have damaged her public image as a worthy Vice President; as she will be perceived as a weak, overemotional woman.
Edit I missed part of your question:
how didn't it "play beautifully for her"? Is this a purely personal regret or a career consideration, too?
It's a personal career consideration for Claire.
Frank says it played beautifully. And he's right. But what Frank means, is that Claire's crying supports the idea that this man's death was a horrifying event. This supports Frank's public opinion on ICO and how they are a threat that must be dealt with.
Therefore, Claire's crying works in favor of Frank, when you consider the race between Frank and Will Conway.
However, the crying also has an effect on Claire's public image. I refer back to my original answer for this: it makes her look weak and unfit to lead the nation.