I have a twin brother and my mother always bemoaned the fact that we were competitive with each other (as all of us siblings were), but it was fun for us in some ways, motivating for us in other ways, but mostly just a non-issue in our minds. And the competition was just about life in general, not primarily academic.
My brother and I were completely different students, though. Come college, we found out that it was likely my brother had some sort of LD. At that point in life he tested as quite bright but not gifted. I had tested as a kid as gifted. I always sailed through school with virtually no effort. My brother, on the other hand, didn't work very hard either but got much more average grades. My parents, in an attempt to encourage my brother, would praise him a lot for the good grades he did get, and I remember just an occasion or two where they rewarded him when he improved significantly on a bad grade. Because I was consistently doing well, however, come report card time I always got a "good job!" but that was it. I'm sure they were trying to minimize the differences between us, and it wasn't really a big deal to me, but I always thought that it wasn't fair that he was rewarded when I wasn't. I guess they were emphasizing effort but I saw it as rewarding him for getting still-lower grades than me.
Also, I always wanted to skip a grade because I was ready for it academically and all my friends were older than me, but it was never an option for us because I had a twin for whom a skip was not appropriate. I think my parents made the right decision here, because it would have been quite awkward to go through school having to explain things. But, I did get to take higher-level classes. It wasn't enough, and I would have appreciated other opportunities, but it was what it was.
I think some things that helped were that we never were in the same class, except for 6th grade band. We didn't have our lockers near each other. We participated in different sports and activities. Our parents never really mentioned our testing scores (although I found out as an adult that my dad had taken my 4th grade ITBS scores to show to people at work when I got 12+ GEs on all the sections but one). And, I think, school wasn't as intense and competitive in general as it is now, so we didn't take it as life-or-death as people tend to do now. We did have some of the same friends, but we mostly ran in our own social circles.
So, anyway, this might not be helpful at all to you, but I thought I'd share our experience. HTH! It's great that you're considering all the elements, so you're on the right track!