Quick poll, let's go.
Let's assume he just does.
Interesting Question for sure.
I have the feeling I represent quite often the minority in certain views about him, but I also do not think he would do it.
Remember the Magnae? People seem to wave it away easily, like, that was just a warning he gave us, as if that didn't take a lot from him.
He went against the wish and order of Xavier, his Master, who he also describes as someone very dangerous, who is not lenient usually and doesn't tolerate other peoples weaknesses, to warn RT. Xavier clearly wanted this to be a surprise and Heinrix went against this - knowing that this will not stay a secret, he even expects to be punished in some sort for this:
"Let them charge me with whatever they want later — I had to make you aware that there is no genuine threat to your life."
He is more afraid of what will happen to RT than to him. If RT acts unwise, Xavier might order her to get killed - or worse - tortured by him. He is clearly afraid of that and this is also where his fear on Janus stems from I think.
You can also have two times (Janus/Black Ship) already told him there is more than duty - but to be honest, even without this, even if hardened, he acts the same in the Magnae, because duty or not, he is already in love with RT.
And while a hardened Heinrix later tortures Emelina, I think RT is different for him, because of this love. Yes, he cares about Emelina too, but not as strongly. And while Heinrix leaves a heretic RT the ending slide says he was killed by RT while pleading to her - he still would not give her up.
Commorragh is the reason he opens up - if he is present or not. Because he realizes how easily he can loose RT and even sees it as another chance for them to at least enjoy the time they have. And you can have his date and him saying he loves RT even before Emelina.
So I think there isn't actually that much between the Magnae and him stating he loves RT.
I am honestly not sure that Xavier has forced Heinrix to do terrible things just to test is loyalty (at least not openly) - I think that came up naturally during his work, making hard decisions like with Rykad Minoris. Heinrix at least doesn't (until further towards the end at least) seem to hate Xavier - even after Emelina look how he talks about him. He sees him as his personal teacher, admires him, looks up to him. He defends Xavier during banter even later with Ulfar. I don't think he would talk like this if he would hate him or be really deeply afraid of him because of what Xavier might have done to him but for sure he is afraid of what Xavier would do to RT.
Would he do it with a stable environment of loved ones and friends? I do agree it means even more for Heinrix because he doesn't have anyone left or any friends really (aside from Emelina) so he is very focused on RT. But even then - he is in love. It would be more difficult for him to go against Xavier's orders because he might be afraid of retribution affecting his loved ones but still he loves RT. I don't think this would lead him to torture her.
As already being said - Heinrix could make it less worse for RT. But Xavier would know this and notice this too. I know it is hypothetical but if Xavier really wants RT to suffer he will make sure of it. So I doubt Heinrix would be able to do anything except for giving her a quick death and then face the consequences.
So no, I don't think he would torture her. He would rather end her quickly but I am not even sure he would be able to do that. He nearly kills himself just over the fact he can't stay with her. He might start a desperate action to have her or both escape but he knows that can't last. The Inquisition can find them. But they can enjoy the short time they have.
I think part of it would depend on the reason and the scope. It's easier to justify a certain amount of suffering for someone's own good, or for the good of the Imperium, vs torturing someone to death.
The Rite of Purification that a Heretical RT can undergo is, in my eyes, clearly torture. It's described in very similar ways to the drukhari inflicted and the RT can think they're dying and beg for mercy. And given that Heinrix has already asked for the RT to repent and is about to leave, I see him as being grateful it's happening. In fact, given that the RT asking Calcazar for help does seem to come out of nowhere, I could definitely see a romanced Heinrix as having strongly hinted that the Rogue Trader needs to seek the aid of the Lord Inquisitor. And he'd be doing that knowing what the consequences of that would be.
(Sidenote: the fact that Calcazar has the tools and is willing to offer the Rite of Purification is interesting to me. I don't imagine he gets many repentant heretics that he thinks worth the effort that could be "salvaged" in such a way. But the rite mentions the stains left by corruption on the soul, and Heinrix mentions he risks similar darkness messing with the chaos cogitator on Kiava Gamma. So I suspect the most common target of that purification is other members of the Inquisition, after the sort of particularly fraught mission that leaves them wanting to scrub their souls in bleach.)
At the Magnae, Calcazar can mention the "varied and fascinating instruments and procedures that expedite candour". And I think that if the encounter at Kiava Gamma where Uralon psykically contacted the RT had gone more sideways, or perhaps if some of the drukhari had said something that made Calcazar genuinely wondering if the RT had contacts with their own faction of drukhari that might foul up his deal with Yremeryss... it could come up that the Rogue Trader could be challenged to "prove" herself under interrogation.
And if that happened, an Act 2 romanced Heinrix would, I feel, consider himself the best choice because with his biomancy there's less chance of going too far and doing permanent physical harm. "The only way she does not survive this is if she's a traitor to humanity who tricked me the whole time" is something Heinrix would cling to.
Which would mess them both up terribly in the medium term, of course. But for the sort of RT who took more 'devil' flirting options, I could see the pair of them patching things up in Commoragh, with some black humoured "well, I knew that 'trial' had to be fake because the hallucinated version of you was doing [x] instead of [y]..." compare-and-contrast conversations as a form of processing.