I allow it against PC Dogs too. The player doesn't lose agency of their character as a consequence of losing a debate any more than they would lose agency if their Dog's arm is broken in a fall. It may be a bit of a challenge to roleplay, but that's no reason to avoid it. If they feel that it is unacceptable they can draw their piece during the discussion anyway and protect themselves at the potential expense of dying / killing. It would take a HECK of an argument to force a Dog to lose a discussion they didn't feel like having.
There's a little bit of a sticky wicket here, and it's one that bears some frank discussion with your players.
Sean's point about player agency is a valid one, but it is not one that a lot of players are familiar or comfortable with. While I too think that it can be an interesting role-playing challenge to have a character's perceptions or beliefs changed as a result of in-game conflict resolution, it has ramifications to player agency in that it removes (or modifies) some aspect of decision-making about the character from the player who created that character.
The "broken arm" aspect is similar only insofar as it might cause the GM to say flat out, "no, you can't do that because your arm is broken." But if the GM is simply applying some situational modification to the fiction or conflict-resolution based on the broken arm, the analogy breaks down. I can
try to shoot you with a broken arm - I'm just likely to fail. But if I have been fooled into
believing that the Steward is telling the truth, I have no plausible reason to then
try to shoot him.
One of the concepts at issue here is in deciding just what "tracks" or "pools" or "resources" are valid for delimiting character "damage," and what effect that "damage" has on the mechanics of the game (or its unfolding fiction). Most people are totally cool with a physical damage "track." Whether you term it Harm or Hit Points or Wounds or whatever, most people have this intuitive understanding that the body can be incrementally damaged, and even that that damage can have ramifications on future attempts to perform some action (i.e. trying to climb a rope with the aforementioned broken arm).
Where people are less cool is when the character resource being degraded is something other than physical, e.g. mental or emotional or social. Or foundational.
MonsterHearts is aces at bringing some of these other kinds of damage "tracks" front-and-center, but it is highly unusual in that regard.
A Dirty World is another pretty good example, because your character's relative levels of Purity/Corruption (for instance) are subject to change or "damage." To some extent, this is a
foundational change to your character - changes or "damage" to it affect not just what you can do but who you are as a person. Though it should be pointed out that even in
A Dirty World, having been "damaged" into being more Corrupt doesn't mean you can't
attempt something that is fueled by Purity - it's just going to be harder to accomplish. Hence, even though your character's core beliefs may have been altered by events, you are still possessed of free will. More on this in a sec.
The problem in an RPG context is that many players come from play-experiences where these other types of damage aren't a thing. People get the concept of a broken arm. People are less clear on the concept of crippling depression, or abject-phobia-inducing shame. And they're super fuzzy on letting the dice dictate a response to temptation. Or altered sexuality. In these cases the oft-heard phrase, "but my character would never do that!" starts to come into the discussion.
In some sense it's a case of different creative agendas at work. Is the player the sole author over the character's thoughts and beliefs? Does the GM have any input? Do the other players? And if so, through what mechanism are these conflicts resolved? Many games have a character attribute that is some sort of intelligence or perceptive acuity stat - does having a low one of these mean I'm ditzy or gullible? Many games have a character attribute that is some sort of willpower stat - does having a low one of these mean I'm easily swayed or weak-willed? How does (or should) that play out in the context of the game? I as a player might be smart as shit and know the NPC is lying - but if my character has INT and WILL as dump stats, he's probably screwed. Is it "right" that I as a player should be able to avoid being penalized for making the decision to play a dumb, wishy-washy character just because I say "my character would never do/believe/fall for that"? Is that equally true if doing so let me place finite character creation resources elsewhere in places that I perceive to be more broadly "useful" (like strength or skill with a gun or just generic bad-assitude)?
The key element here is one of intent versus consequence, and I'm afraid to say there's no "right" way to do it. It all boils down to what your system supports and what your players are comfortable with, and the only way to know that is to discuss it openly. "Can I
make you believe/hate/trust/like/fuck me just by rolling dice?" Or "Can I
make you believe/hate/like/trust/fuck
yourself just by rolling dice?" Or can I just make it so hard for you to do anything else (i.e. by applying so much "damage") that you'll eventually either give up or die trying?
This is where the concept of free will comes back in. I think it's perfectly OK to say, "Dude, you got hammered in an emotional conflict with the Steward. Any further action you take against him until you get your mojo back is going to be penalized because you have this nagging doubt - by the King, he's so well versed in the Scriptures that he might just be right!" This stops short of saying, "Dude, you can't take any action against the Steward because you believe he's right." This difference is subtle but important - one way preserves sole player authorship over the character's beliefs (though subject to "damage") and the other does not.
In the context of DitV, what you're talking about is setting the stakes. As the player or as the GM, I think it's totally cool to take a stake right off the table if it crosses that line of player (or GM in the case of NPCs) agency. So it's totally okay for the GM to respond to "I want to convince Sister Abigail that smoking is a sin and make her give it up forever!" with, "Nope, not a valid stake. You can convince her to stop smoking right now, maybe even make her
question her beliefs, but she is who she is." Similarly, it's totally cool for a player to respond to the GM's "If you fail/give, you blow your vows and totally have kinky, nasty sex with Sister Abigail the Smoker" with, "Nope, not a valid stake. But I will be all hot and bothered and distracted and inwardly shameful about it."
Setting the stakes is a two way street. It all comes down to what you and your players are comfortable with.
In the case of DitV in particular, I think a lot of times people (players and GMs alike) leap way too far with the stakes right off the bat. They see a problem (Sister Abigail is smoking, which is a sin) and want to jump immediately to the "solution" (convince Sister Abigail not to sin). I also think a good rule of thumb is that you can stop or change what someone is
doing but not who they
are. So you can stick your gun in Sister Abigail's face and threaten to blow her brains out unless she stubs out that blunt pronto. But as soon as you're out of sight, she's gonna blaze up again if it suits her fancy.
Does this help?
P.S. - none of this deals with the really interesting question - why is Sister Abigail smoking in the first place? Is it just because she's a libertine with no regard for the King's moral authority? Or is smoking ditch-weed the only thing that relieves the chronic pain of the hard labor forced upon her by the wicked, greedy Steward?