Actually I think you make a very good point here. Jaqen didn't do any killing until he "owed" three lives to the Many-Faced God. I'm working from memory here but I believe he told her he wouldn't help her kill anymore than the three lives he owed her. He was refusing to help her escape from Harrenhal because it meant killing more people and that would have been more than three (at the time he only owed her one more life). It wasn't until she named Jaqen as the third life did he relent and kill more people. His life was threatened (like you pointed out) and so he was able to do what he had to do to save himself, and in the process save Arya as well.
This earlier post got me thinking about an inconsistency with The Faceless Men - Or at least with Jaqen H'gar that makes me wonder if he's fully affiliated with the Temple of Black and White.
The Faceless Men in the house of black and white are assassins that only kill when someone gives them a name in the Temple. Arya was reprimanded (by losing her sight early) for killing Dareon for her own purposes.
Jaqen seems to stretch the rules by giving Arya three names to make up for the three lives she spared. In Arya's training in the house of black and white, they never say anything about how someone who spares a life is allowed to select a death - particularly if that name is said outside The Temple.
In addition, when Arya names Jaqen -- as opposed to killing himself, Jaqen kills numerous guards. This seems like a clear violation of the rules in the Temple of Black and White. That is, unless the "death" of Jaqen H'gar is him changing his face to look like someone else. Still, he kills an awful lot of unnamed people before changing his face.
When Jaqen kills Pate, it's so he can replace Pate and infiltrate the Citadel. Its doubtful that Pate's name has been said in The House of Black and White. It seems to me that Jaqen is acting more like a special agent than a servant of The House of Black and White.
I wonder if not all people who are trained in The House of Black and White remain to serve. This would certainly allow Arya to have a nice out. Maybe Jaqen H'gar thought The House of Black and White was a good place for Arya to get her training.
Of course, one problem with this theory, is that The House of Black and White views death as something religious. Still, I doubt they'd be thrilled to train people who go about killing for other purposes...