Would Carlisle/Alphard work? (Platonic or romantic)
... curse tumblr for I had drafted my reply to you. ALAS.
Carlisle is not for Alphard
Alphard is an extremely cynical person who admires Tom Riddle for his strength and infallibility. Tom is the most extraordinary person in the room at any given time, and always true to himself. As far as Alphard is concerned Tom is a demigod among men, the sort of natural force who doesn't live by the same rules the rest of us do and wanting him to change is the last thing on Alphard's mind.
Would he admit this to Tom's face, never, Tom has enough of an ego. Alphard will call him a lunatic and ridiculous, and mean every syllable. Did he fall in love with a violent lunatic with impure blood who was beating up not just Alphard, but his closest relatives and all his friends in school, also yes.
It's the whole package of Tom that makes him appeal to Alphard, from the physical beauty to the uncompromising personality, to the way he can't ever be fully predicted, and the tragically romantic backstory. Being in love with him is just a point of fact for Alphard at this point.
Even becoming Lord Voldemort is something Tom never claims is anything but what it is, and while Alphard is horrified and heartbroken Tom remains the person he always was. Readers of The Man Who Would Be King will remember Alphard lasted one week before being married to Tom again.
Carlisle, by contrast, while unbelievably beautiful and just as extraordinary, is a man who has made self-delusion a cornerstone of his life. He loves his family and wants them to care about human life as much as he does, so he'll give them little nudges like going to their victims' funerals or have family votes where thankfully the majority voted against killing an innocent girl, and not think about what it says about Edward that he killed people for pleasure for four years because- well, he came back.
And he walks around talking about how great, how humane, how wonderful his family and their way of life is. While living among humans, thereby risking the deaths of innocents for no reason other than "it's our lifestyle!" (and the even worse, underlying reason of "if they don't live with humans they might forget humans aren't food...")
Loss of control isn't even a hypothetical, this happens to the Cullens semi-frequently.
Alphard would think he's a fool and a killer by proxy, and despise and pity him. To him, Carlisle is easily worse than Voldemort.
Alphard is not for Carlisle
The trouble with Alphard is that he is what Caius would be if Caius was worse. He's mean, he's judgmental, and he's cynical, all qualities Caius shares only Alphard is somehow worse. He's just so mean.
More troubling yet, he is very principled and harsh on himself but lives cease to matter to him where his loved ones are concerned. Had Aro said "Here is my Horcrux, it's a fifteen-year-old Aro who must be fed a soul to gain a body" Carlisle would have pressured him to either repair his soul, and left when Aro didn't do so. Alphard, by contrast, "Ope, guess we're finding him a soul then."
Alphard is a very ruthless person, he may be principled but should his line of reasoning lead him to murder being the solution to a problem a loved one is having then murder it is.
Alphard also reacts to Tom becoming Voldemort much the same way he would infidelity, as it's not really the suffering Tom inflicted that bothers him but the betrayal of his own character as Alphard knew it (and he'd have had a much harder time getting past actual infidelity. That would have been a crisis). His faith is restored because he sees enough of the goodness he fell in love with. His niece Bellatrix is much the same, of sure she's done bad things, Alphard is intellectually aware of this fact. It's getting hard to deny that she probably has tortured and killed people, and delights in it. Well, have you considered the fact that she's precious and perfect?
Andromeda's marriage to Ted is on par with Tom and Bellatrix's life choices in that Alphard's not thrilled with it, but he can look past it because he loves her that much.
To Carlisle this man is genuinely insane and terrifying. Carlisle can move past his friends killing to live because it's what they've always known and he sees the good in them in spite of that. Alphard would frighten him, there is plenty good in him but Carlisle would correctly put together that the man is one line of reasoning away from killing anybody at all.
Carlisle stays as far out of his way as he can, and warns others to keep their distance from this one.
Can these two even be in a room together?
I think if they meet in the library and only talk about books, they'll have a grand time. Just don't let them talk about anything personal, at all.