There's no "might" to it. Mudd WAS cooperating with the Klingons and would have betrayed both just to save his own neck. You don't put yourself and your crew in jeopardy just to appease a higher sense of morality.
So what? Here is a universal truth: Torture breaks
everyone. That's why it's not a reliable information gathering technique (even if entertainment regularly tries to sell it you as that): Under torture people do
everything and say
anything to make it stop. A "confession" under torture is worthless, the same as the "information" gathered - just ask like half of the inmates of Guantanamo Bay. If you really want to dig further into this subject, might I recommend you this John Oliver video, which presents the fact very informative and in an actually watchable way:
That being said: Tuggin Mudd along wouldn't have put Lorca or his crew in any more jeopardy than they originally were. In fact, even less, six eyes see more than four eyes. And Mudds "information" would have been worthless anyway: What did he do? Hid a
listening device? In a
torture cell? As if there aren't buttloads of them in the walls anyway.
That Mudd was a criminal worth putting before a judge doesn't change this or takes away his humanity.
But apparently this is what people want. Most people seem to love Lorca, and they had a poll on leaving Mudd behind on After Trek. Something like 75% of people voted that it was a right call and then the host joked about it. People want to see tough guy arseholes.
No I don't get it and I don't want that sort of Star Trek, but the truth is people like us are in the minority.
Sadly, this succesfully appeals t our lower instincts: Somebody wronged us, now we can pay them back
tenfold. Just ask how many people would like to have some sort of come-uppance to their schoolyard-bully. Humans do seem to have a knack fo pure vengeance, and the usual Hollywod entertainment surely cultivates and helps to grow it.
This is one of the things where Star Trek in the past was special. TOS and TNG especially, but even during VOY or ENT the characters worked toward an agreement and reconciliation (see the Xindi-arc) instead of the usual "defeating" of the "other" guys.
We will see how DIS develops. I think it might surprise us in the future, and the war with the klingons doesn't look like one to be "won", but one to be "settled" in some sort of way. And since my personal pet theory is that Lorca will turn out to be the Big-name-actor-British-surprise-villain (see "Wonder Woman" for latest example) of the first season, I give them more leeway to act in "bad faith" when he's "alone in the dark" and nobody there to watch him.
But yeah, the kind of following his obvious immoral ways and choices have
is kind of frightening, although not exactly
surprising, given the current political climate all over the world.