Vetinari has always had moments of over-confidence - especially in "Men at Arms".
Sadly, I don't think we'd see a Lewton cameo in a Discworld book. Anyone who hasn't played
Discworld Noir for whatever reason (lack of funds? Game not available in the country? Lack of interest? Etc.) wouldn't have a clue who he was.
If Lewton did make it into a Discworld book, there would need to be two things:
1. Firstly, it can't just be a passing "Hi and goodbye" scene. New readers would need to be introduced to who he is.
2. Secondly - and more importantly - there needs to be a reason for him to be there. The Watch already does most of the job that Lewton does. So why is he even there, other than for hardcore Discworld fans like us?
3. Thirdly, if Lewton is there, that means that law and order has broken down (as it usually does in the Noir genre). When the police are on the take, or even actively working with criminals instead of against them, that creates a vacuum for people like Lewton (or Sam Spade, or Philip Marlowe, etc.) to step in and save the day. But if the Watch ever deteriorated to that extent, I'm pretty sure that Sam Vimes would have something to say about that!
The only way Lewton could be a character in A-M would be before the Watch got organised (i.e. either before "Guards! Guards!", or between G!G! and MAA). Besides, Sam Vimes is himself a law-and-order Lewton figure.

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Finally, I'm not sure if a Discworld novel
can be written from the sole POV of the Patrician. The Patrician has far too much responsibility and power to be the protagonist of a novel. Where's the challenge? What's the goal he's working towards, the setbacks, the hardships, the decisions he makes? He has far too much of a sense of duty towards A-M to chuck it all and go gallivanting all over the place, like Vimes or Moist.
Besides, even before Vetinari was the Patrician, he was always a subtle character who worked from behind the scenes (see
Night Watch).
If it was me, I would write a Discworld prequel (possibly set at some time between the death of Lord Winder and the death of Lord Snapcase). That would make Vetinari sometime in his early 20s and a graduate of the Guild of Assassins.
As for plot - Snapcase would be fearful of the Guild generally and of Vetinari specifically. Perhaps he decides to send Vetinari off somewhere dangerous and remote (Überwald?), as a way to get rid of him. Perhaps this is where Vetinari meets and forms a wary alliance with a certain Überwaldean noblewoman. Hmm?
It is also possible that when the Patricianship changed hands, and Snapcase demanded files be created on "up and coming young men in the Guild of Assassins", that would have been a
very good time for Vetinari to embark on the leisurely coming-of-age Grand Sneer, customary to young men of his social class. If Snapcase was seeking to establish a shortlist of candidates to ascertain the identity of the Assassin who "got rid of" his predecessor, it is highly likely that Vetinari would have been offered the same sort of grateful advancement as was offered to John Keel. This would have made it an ideal time to leave the city for a few years and exploit his aunt's Disc-wide social and business connections.
So yes - given those circumstances, I can quite see a novel centred around Vetinari in his earlier years. Aristocratic, well-bred, dangerous, but broke and homeless. What would his goals be? To find a new home? To be able to return to A-M? To discredit or get rid of Snapcase? But he would be far too law-abiding to consider that last one.
Perhaps, after a few years, the Assassins' Guild is fed up with Snapcase's incompetence and puts a price on his head. Perhaps Snapcase becomes like Winder, but better armed and more competent. Perhaps the Guild tells Vetinari that the price of his return to A-M would be Snapcase's inhumation. But Vetinari, with his better connections (Lady Margolotta?) and his wealth of experience, surprises everyone by not only killing Snapcase but also assuming the Patricianship at once and putting an end to the chaos.
Now, if there was a Discworld novel set around Vetinari,
that is the kind of novel I'd like to read!

What do you think?