i am not going to punch him.
The thing about Red Prince is that he is a man of passion, which is mentioned many time.
And the trigger to the romance is mentioning the jealousy and light that fire in him.
The problem is that the passion died in the epilogue (for some reason).
Let's look at the 3 choices you are offered and assuming that the writer actually play some dating sim, this is what each choice will lead to.
1) Accept the slave thing and say it is wonderful
Bad end -> The character has reverted back to Act 2 romance status with the Red Prince. The Red Prince only cared about the player because the player mentioned it and asked for it. With Sadha absence, he was able to give the player the care he asked but with her around, the player pretty much loses.
2) Walk away
Neutral end -> The Red prince didn't even show care of pursuing the interest so to walk away, the player is also cutting off the romance.
3) Punch him
Neutral end or Good end
By punching him, the player is now attempting to develop the relationship he/she has with Red Prince. Remember that Red Prince is a passionate person who always long for challenges and excitement with a hint of love.
At this point, what the game needs is an option to start making demand for the love (or choose to drop it for the neutral route). This is why the punch is required especially with someone like Red Prince who assume he already has everything in control. Before you make demand with him, you must break his cool and as mentioned by the narrator, he won't even expect it. You must remove yourself from the status of someone asking for the love and move to someone demanding the love.
And when it comes to making demand, the player must also reignite the passion the Red Prince has in order to force him to take you in.
So right after he says "Next time, use your words", they player should be given an option to be the one to force a kiss on the Red Prince follow by this kind of line.
"If it takes for everything to be shrouded in darkness for you to be my light again then I'll give you one hell of an oblivion. Shall we see how deep this water runs?"
If they ever fix the epilogue, that's how the Red Prince route should be to stay 100% true to his character and give you the best ending where you turn Sadha into a sidehoe.
Shame the writer doesn't do any of that.