King Saul appears to have wanted to die rather than fall into the hands of the Philistines, because of what they would have done to him -- and how that would have destroyed the morale of the Jewish people. (And that's not halachically so clear either.) But that's not your question.
Achitofel hanged himself so his family would inherit his estate, whereas the estate of someone executed for treason is given to the king. That is definitely not halachically condoned, and isn't your question either.
The rabbis of medieval France debated whether someone may commit suicide rather than being tortured into converting to Christianity. Still not your question.
"Family's honor intact"? We view life as worth more than that. Furthermore, if someone truly deserves capital punishment, we view that as part of their atonement process.
Rabbi Akiva Eiger's commentary to Makkot 5a:
With regards to monetary debts, even if there were no witnesses and the courts couldn't force him to pay, if he himself knows that in truth he owes the money, he is obligated to pay ... [whereas] with regards to the death penalty -- suppose a person knows in all truth that he killed someone -- he is still not "good as dead"; there is no obligation whatsoever for him to kill himself. There is no obligation whatsoever until the courts render a sentencing.
Philosophically, the message behind "judicial suicide" would be to tell people "you should be so ashamed of yourselves for betraying the king, that you'd want to kill yourself!" Or better yet: "no one ever wants to betray the king. [We quietly sweep those who do under the rug by getting them to commit suicide]." By leaving this punishment in the hands of the king -- and, in fact, at the discretion of the king, and making it public, we are saying publicly "society cannot exist without authority." As Lord Sacks, the previous Chief Rabbi of the United Kingdom, points out, we are a "guilt culture", not a "shame culture."
Just from a technical perspective, if someone commits treason, the king can decide whether to have them beheaded or let them off the hook (Maimonides Laws of Kings and their Wars 3:8); so it would be quite silly for someone to commit suicide when the king might pardon them anyhow.