Yes, that's right. Both are correct. The sentence uses the personification. The causee is the trash, and 部屋のゴミをもう一泊させる "they(= the hotel) let the trash of the room stay one more night" actually means "they keep the trash(= they don't throw the trash away) for one more night".
Can you give us your translation of the whole sentence? This is a general idea and not just to you, but we often see the cases that "learners think they understand the meaning of a sentence" is not the same as "their understanding is correct". I believe it's useful for you to confirm whether you really understand it correctly or not.