He does like us, so he is unlikely to press us further into fighting the magistrate if we can pass this test. We should ask
@Maugan Ra for an etiquette check on whether or not talking about the family trade would help or hurt when we make a suggestion.
Falling back on your family's reputation and known methods is neither inherently positive or negative. It is, in fact, what is
expected, and that can be either good or bad.
How your suggestion would be received would then depend on what Kitsu Ami thinks of the Soshi family and the Scorpion as a whole, rather than depending on what she thinks of you as an individual.
@Maugan Ra
Would suggesting asking for help from a Kitsuki be considered rude? Probably with something included that the Kitsuki method, for all of its unorthodoxy, works. This is something our Clan can likely and unfortunately attest to.
@gman391 is correct here - advising that the Lion seek the aid of the Dragon here would be deeply unlikely to go over well. The Lion are the most traditional Clan of all, and the Kitsu are the keepers of their sacred traditions and priests of their honored ancestors.
In the eyes of the Lion, the Kitsuki are a young family (only just about three hundred years old) full of strange ideas. They are unorthodox, come from an unorthodox Clan, and practice a form of investigation which inherently relies on doubting a samurai's given word and favoring objective truth over what is the most honourable course of action.
You can get away with suggesting that she ask the Scorpion for help
because you are a Scorpion - demonstrating pride and a confidence in your own family's techniques is fully expected and laudable among samurai. If you call upon the fact that your Clan is officially appointed by the Emperor to do this sort of thing you even win bonus points, because the Lion are all about doing your imperiallay-bestowed duty.
Recommending that she consult the Kitsuki, on the other hand, is liable to get you challenged to a duel unless you are
very careful in how you word it.