So which roll should be made to hide pain so great it would make even the strongest man squirm?
As always, things not covered explicitly by the rules are down to the DM, but here are my thoughts on the matter:
Constitution, measuring endurance (PHB p.171)
A con save would allow a character to resist the torture, from a strength of body point of view, but would not allow them to hide the pain they are feeling which is quite real and will have clear physiological effects. Constitution measures the ability to endure physical hardship, not fool anyone. A con save would allow them not to "break", but not to lie convincingly about what they are feeling.
Wisdom, measuring perception and insight (PHB p.171)
I agree with you that a wis save is not right. It is not a measure of will power. The reason it is used to save against fear and charm effects is having the wisdom, the insight, to see it for the magic it is, or for the folly of panicing in the face of disaster or responding to the feelings of love etc. to someone you didn't moments before. It is not force of will and would not help resist torture.
Charisma, measuring force of personality (PHB p.171)
A cha save, however, would serve to resist the torture, looking at it from a force of personality perspective. It would be down to the DM to decide if a successful save meant that the character showed the pain or not, but I would say not as this is not the purpose of saving throws: they are resistance to adversity not active abilities. The fact that the character needs to make the save means that the pain is quite real real and so great it threatens to "break" them. It is avoiding "breaking" and spilling the beans that they are saving against.
Deception (PHB p.178)
Your Charisma (Deception) check determines whether you can convincingly hide the truth, either verbally or through your actions. This deception can encompass everything from misleading others through ambiguity to telling outright lies.
What would really allow the character to hide the pain, or even fool/tell a lie to satisfy the torturer, would be a cha(deception) roll. If the character has "broken" (failed their save) then they have lost the ability to choose to lie or fool the torturer, but otherwise they can pretend that the torture has no real effect depending on what they are trying to achieve. They could pretend to break and lie about whatever the torturer is asking (if anything).
So I would ask two rolls of the character to do what you ask, one to save against "breaking" and one to deceive their torturer into believing that they feel no pain.
ps. on a side note I have often thought that cha saves often are vastly under-rated for the exact reasons given above. Charisma is the closest to a will power stat that D&D has.