Neither of the gyms I went to had an official "diet" or did any weighing, they just stressed healthy eating in general and encouraged us to bring healthy snacks. If it was a one time thing and wasn't super healthy, it's not like anyone would say anything, but if someone was consistently bringing soda, candy and chips for their lunch in the summer (we didn't have a break during the school year because practices were 4.5 hours, although we could go eat or get a drink quickly while we were waiting for a turn), then I'm sure they probably would have talked to the parents, if it was a younger kid, or the gymnast herself, if she was older.
As a coach, I really wouldn't want to be responsible for monitoring it. If I felt it was necessary, then bringing someone in is what I would choose, because I would want it to be a less loaded third party interaction.
My problem with BMI is that while it has some use for rough measurements, it it often not a great measure for many athletes, and particularly for the gymnast body types as it doesn't take into account things like a much higher amount of muscle and broad shoulders/upper body body type that is characteristic of advanced gymnasts. I probably would not want someone who was unfamiliar with athletes and specifically gymnastics to be just checking their BMI and giving them recommendations because I think there would be potential for inaccuracy. There are other measurements that can be used in conjunction with BMI to give a more accurate picture. I would not want a gymnast to get a skewed image of her body type because it is being compared using a formula that doesn't really have a lot of room for body adaptations. On the other hand I think if used sensibly, it is a tool that can contribute to assessing overall health.