Bandit was a very good patient and rarely fussed in the crate even though she had not often been crated before her back problems. Bogart, however, was VERY vocal I actually think it's a good sign that he lays on his back. Mine did not want to lay on their backs for a while after having back issues so maybe that means he's feeling good? IDK, but I agree that startling him would be much worse. I'd just let him be.
The stroller is really a help. I wish I'd known about strollers when Amanda and Bandit had problems. It would have really helped. I did all the lifting with both of them and, ouch, did my back know it. You'll love using the stroller and he probably will, too.