I tried looking at the transcripts, relevant scenes and expressions. From a transcript:
Han: This is my museum. It is difficult to associate these horrors
with the proud civilizations that created them. Sparta, Rome, the
Knights of Europe, the samurai. They worshipped strength, because it
is strength that makes all other values possible. Nothing survives
without it. Who knows what delicate wonders have died out of the world
for want of the strength to survive.
Roper: What's this?
Han: Oh...a souvenir.
Han: Up here.
Roper: A guillotine? Uh, no thanks, this is the only angle I care to
see it from.
Han: If you please.
Roper: You mean you want me to put my head on that thing?
Han: An act of faith.
Roper: Um, I'm a man of little faith, Mr. Han.
Han: Very few people can be totally ruthless. It isn't easy. It takes
more strength than you might believe.
Roper: Now you've got eight more.
Han: Than there is a point you will not go beyond.
Roper: Faked out again.
Han presents his views of strength and ruthlessness. He probably expects his associates to be as such. He also checks if they are willing to put their faith in him. Subordination possibility, perhaps. Or maybe to check if they have a mind of their own, and the ability to reason instead of blindly following orders. Here, he probably notes that Roper has compassion, and would be willing to risk the punishment of insubordination if he was asked to be ruthless.
Han: Just lost drunken men who no longer care where they find
themselves each morning.
You left some rather sizeable debts in America.
Roper: Dinner's Club hasn't called in my card, yet.
Han: There were some questions which I was forced to ask. I got no
Roper: And you want me to join this?
Williams: There are certain realities. I want us to have a clear
Roper: No, there's no misunderstanding between us.
Han also shows that he's possibly done a background check of Roper. Then he shows he's willing to be ruthless with people (Williams) who disobey him. Roper also looks at the two guards near Williams, and notices their expressionless faces.
Han: Good morning Mr. Roper. We have been waiting for you.
Roper: What's going on?
Han: Would you be good enough to participate in this morning's
What are you gonna do to him?
Han: Not me, Mr. Roper, you......Bolo.
Roper: Like you said, there's a point I won't go beyond.
Han: I was right about you. We shall strive to be worthy of your sense
of grandeur. I will find someone with whom you can fight.....Bolo!
Roper is escorted to the ground by a few guards and told to stand at a spot. This is something a person like him wouldn't like. To be treated like this.
Han seems somewhat respectful even now. But he mostly always is.
Bolo has a strange smirk when cutting off the ropes binding Lee's wrists. The kind of smirk he has when he enjoys seeing people in trouble. Probably the kind of ruthlessness that comes with being under Han.
Roper approaches Lee and looks at him with a peering, slightly inquiring expression that indicates: "What's going on here? Did you do something that makes you guilty?". The slight tilt of the head could also indicate asking: "Do you want to fight?". Roper does not seem to have a doubt that he could take on Lee. But he does seem to wonder if it is right to do so in such a situation. While being controlled by Han. Especially after seeing what Han did to Williams, Roper understands that Han can kill anyone. This kind of ruthless behaviour (and perhaps the wanton/vain display of ruthlessness) goes against his conscience. He came to fight in a tournament. This was no longer a tournament. It was an execution. He didn't want to be an executioner. Besides, in all his interactions with Han, he was being asked to do or accept things that were below his sense of dignity and morality.
Lee stares back with another slight head tilt and an expression that seems to indicate: "If I have to fight, I will...and I'm not going to hold back. You don't have to do this".
Even after Roper refuses, Han is still smiling and respectful. Why he asks Bolo to fight Roper, I don't know. Maybe because he's confident that Bolo will kill Roper, and since Roper, who has seen so much of Han's secrets wouldn't be a threat after being eliminated. Perhaps also to show the others that even Roper's insubordination wouldn't go unpunished. The plan is probably that after Roper is killed, Lee would be next. Doing it in front of everyone else cements Han's reputation of being ruthless. Han was cold toward even O'Hara because O'Hara ignored Han twice during the fight with Lee.
I have made a lot of assumptions here. Unfortunately, there's no other transcript I could find, which offers more insight into what the writer had in mind.