I'm a bit confused by the delayed puberty comment. I don't know what Tanner stage so can't comment on that. Puberty to me - means hair under armpits, voice change, pimples, etc. That's how I knew for my boys - and that was early, before 15 (I think you said in an earlier question).
One of my kids is short in comparison to siblings and also cousins, and also friends. He's the shortest on his teams. He may be done growing. We have one relative who was shorter than most - but everyone else is tall. He may just take after him, or he may have a growth spurt coming. No one here is concerned about it - including our kid.
I am not very tall myself, and I have a BIL who is my height and I have a very good friend whose husband is my height so men can be short. It happens. Your son will fill out. These men, hit puberty. I don't know if testosterone shots will make your son grow.
My BIL - everyone else in his family is taller than him. All his sisters, are taller than him. He just happens to be shorter.
I have a very thin son, who is like a bean stalk. He eats non stop but you would never know it. They come in all sizes and shapes as teens. At 15, on our teams, the kids grow at huge different rates. Over the summer, some kids shot up and I barely recognized them. One of my good friends at high school had a huge growth spurt in 12th grade.
I don't know why you are as concerned as you are. One of my son's good friends (the thin tall one) is very small and looks really young. He gets some teasing - but trust me, they all get teased for something. The really tall ones, get teased too. The ones with bad skin get teased. The ones with funny hair, get teased. They are all getting teased at this stage.
I would wait - as his doctor said. I really would let it go for now. I don't have experience - other than having sons who come in different shapes and sizes and all their friends are the same. You would think some are men (some have full beards!) and some still look like little boys. It's all ok. They catch up :)
ETA: I do apologize because I know you're looking for people with experience with the injections and I answered without having that so just skip over my answer if it's not helpful. I just wanted to say that if your teen is concerned, he needn't be necessarily. If his doctor isn't - I'd wait (is all I'm saying). I personally think if you jump the gun (before the doctor is concerned) you might hurt his self esteem (make him think something is the matter with him). I know it would my boys.