We didn't transition Aidyn into a big bed until he was 2yrs 9months old. And then it took a good month+ of hard work - lying on the floor for hours each night next to his bed so he would stay in there. (and now months later things have gone backwards again, and he is spending every night in our bed!)
I think a car bed is a great idea though, as it has some similarities to a cot, with the sides etc. I agree that you should set it up a few months before the new bubs arrive, and try to get him gradually used to it... All the best!