Is “in room 431” better? I agree a change would be better, but would choose to use ‘from’ (because they are no longer ‘in’ the original rooms).
Is “for the room change” necessary? No. It’s optional.
Is "in the morning on October 10 better or both OK? No 'On the morning is correct. The general rule applies here: use ‘in’ for the month only but use ‘on’ for a specific day in the month (October 10).
Is “At the time” necessary? I believe so. Otherwise it sounds as if Saito is asking whether the cases should be placed outside the doors for the duration of the stay.
Just some minor points in this passage:
Scene 2 Marking room numbers on the baggage (At the bell captain’s desk)
Saito: Excuse me. Could you tell me where our baggage is?
Bell captain: It’s right over there.
Saito: Some people have changed their rooms. I’d like to give you our latest rooming list.
Bell captain: All right. What are the changes?
Saito: The Takahashis from room 431 have changed rooms with Takagi and Yamaguchi from room 478.
Bell captain: Oops! We’ve just havehad the baggage carried to the fourth floor.
Saito: No problem, I’ll go and handle it.
Bell captain: Oh, don’t bother yourself about it, Ma’am. That’s our job and we’ll take care of it.
Saito: Thank you very much. Then I’ll go and tell them there is nothing to worry about for the room change. By the way, I believe porterage can be paid with the voucher, is that correct?
Bell captain: Yes, it can. And will it be all right to bring your baggage down to the lobby at 6:30 on the morning of October 10?
Saito: Well, we leave at 7:00. Will there be enough time?
Bell captain: Yes, I think so. We have a couple of groups leaving that day, but thirty minutes should be sufficient.
Saito: Okay. Just one more question. At the time, should we put our suitcases outside or inside the doors?
Bell captain: Please leave them inside the doors. Porters will go and bring them down in due time.
Saito: I see. Thank you.