Please before you do anything else, go "up the chain" in the school and make sure the principal and anyone else who needs to be OK with this decision is OK with it.
It may be overkill, it may be totally fine, especially if you're just going on the trip as well as opposed to being a chaperone (you didn't specify). But it's possible that various aspects of the trip is being insured, including the rooms. If 2 people are in the room that are not part of the school trip's manifest, and something happens in the room then it becomes an issue as to who's responsible. I'm sure you and your family are all responsible people and wouldn't jeopardize the trip, but if the trip IS insured, then the school is probably under contract to adhere to certain policies, and one of those policies might be "no one else beyond who's on the school trip is allowed to be in a room booked through the school." And the teacher who's totally cool with it might not be aware of those policies and just think it's great that so many people want to share the experience of the trip. Know the policies that the school must adhere to before making a plan like that; I'd hate to know that your hopes were dashed by not investigating. And I can't imagine anyone would fault you for wanting to be sure it's OK to bring more family on the trip, even if they're not on the bus but just in the room.