Not my experience but that of a friend. Her baby was born with RDS after being induced at close to 38 weeks (although by the looks of her daughter the dates were out and she was less that 37 weeks). She had to be transferred to a different hospital as the local hospital had no beds for a baby who was that ill. I don't know full details of the treatment but she was home less than 2 weeks later and has had no long lasting effects. Another friend more recently had a baby born at 36+5 weeks gestation. He had jittery blood sugar levels, but no RDS, and only needed 3 nights in special care before he was allowed home. At 36 weeks you will not be offered steroid shots, as that would not prevent RDS, it would just help strengthen the lungs a little and is more appropriate for very premature babies.

I think your reasons for looking for an early induction are very understandable. From what you have said about your OB he sounds great, and whilst he is unlikely to give you an early date right now because of the risks, I feel confident that if your mental health required you to have an induction closer to the time it is something he would give consideration to at that time.