Honor is a good shot and beyond that she has zero reasons to believe that Houseman himself would be any good at duelling.
Well yes, but it would just have been such delightful twist if Houseman had risen from the floor and, shakily, said "I challenge you."
Not that a liberal strawman character,
even in a dueling culture, would ever, ever do that.
It's one of the reasons I find Weber's choice to 'integrate' dueling customs into the Honorverse to be so damn
dumb. Because his characters mostly don't act like 18th century nobles (or commoners) who live in an aristocratic dueling culture run by a hereditary oligarchy. They act like 20th century citizens of a democracy, who
just so happen to have a government shaped otherwise, oh and a dueling culture as an add-on.
The second-order implications of dueling culture and other such things are just...
dropped... whenever they'd be inconvenient to the plot.
It's like someone else said. Weber, in this series,
likes hereditary monarchy and aristocratic privilege and thinks it would be just grand if insulted people could legally challenge the insulter to a pistol duel. So he includes those things in his setting. But, because he does this out of a
liking for these institutions, he doesn't think through the implications, or the ways they would shape society to make it noticeably different from the late 20th century democracy under late stage capitalism that he
personally lived in as his formative experiences.
Houseman is presented as being mega-cowardly, especially as he's issuing orders he doesn't have the right to give (he's not the Manticoran ambassador so his legal position is iffy at best). The one thing he could have done is dragged Honor through the courts but whilst that would hit Honor pretty hard it'd also put Houseman in a bad light.
I mean, arguably. On the other hand, Houseman's lawyer could make a very good case that any testimony about what Houseman said before Honor slapped him was immaterial and not admissible as evidence. Houseman's final words were "I'll have your commission! I have friends in high places, and I'll—"
That is not normally grounds for assault.
I don't think a Manticoran court, many of whom
identify with the kind of "high places" in question, would want to set the precedent that it was.
Houseman probably couldn't make a case for "disobeying lawful orders," but he could damn sure make a case for assault.
Houseman's hysteria is pretty darn caricature-ish since Houseman could and indeed did stay off the ships when combat started. His argument about risks only works if Masadans are prepared to breach the Eridani Edict by firing on the planet and considering the near universal cultural importance of the Edict and Manticore's own cultural quirk on last man stands (Saganami is explicitly Manticore's top military hero) that's not really a great argument to make irrespective of legal merits.
To be fair, Masada has bombarded planetary targets in the past, and are a bunch of murderous maniacs, so Houseman is right to be afraid of the consequences of a Masadan victory.
On the other hand, Weber explicitly wrote Houseman as believing the Masadans can be reasoned, negotiated, and traded with, so this isn't
consistent. See previous comments about Weber's tendency to attribute to his characters and societies various traits he likes or dislikes, without stopping to think through the logical implications of those traits- the vices of their virtues, or the virtues of their vices.