Nevermind the dog. I wouldn't trust the friend.
She's only recently adopted the dog, therefore I'd assume she's not had it long enough to get used to its temperament in different situations.
Is she a new dog owner, or does she have lots of experience? She's very casual about taking it on a long car trip, a long day out with its new owners friends, and saying Oh it can go on the back seat. That sounds like a very long day out for a new dog. That suggests to me that she doesn't have much experience. If she did, that might be a different matter because you could be confident that she knew what she was doing and how to control the dog.
As it is, I wouldn't be surprised if she hasn't thought this through and I'd be asking her how she secures the dog in the car normally.
I'm confused about who is driving, you, the friend or another friend? If dog owner is not the driver, I'd be voicing concerns to the actual driver. I doubt if they will be thrilled.
If the friend won't be moved, I'd probably Get the train.