I like Mel's advice.
You could use this as a very effective teaching moment. Let her know you think she's doing well and you'd like her to practice with you on how to approach situations when you need to talk to an authority figure.
"Honey, you're doing really well and it was nice of you to let the older kids have your place in the meet. I'd like to help you work on some skills, if you'll bear with me this might be fun!"
When your coach comes up and asks you if you'll move aside to let someone who didn't do as well as you to race it might help if you asked him if there's anything you are doing wrong and if so can he help you work on it. SO let's practice how that conversation might go. I'll by you and you be coach".
Coach:
"Hey kid, I need you to let Mary and Jenny race on the varsity for this race".
Kid (mom).
Figure out beforehand how to ask this question where it's both showing him the proper respect but also sort of putting him in the hot seat because he's not giving her the proper respect and following his rules.
"Well Coach, I thought I did okay, is it something I did wrong? Is it something I can work on so that when I do place well again I'll get to race in the right position? What can I do to be better and not have to move back to JV"?
Coach
"Um, well, um, I think you did a fine job but Mary and Jenny are older and they won't be able to race after this season and they'll be gone next year"..blah blah blah
Then turn the tables and let her be the kid and you be the coach.
Practicing several times until she's confident she won't freeze up and stutter and hang her head and say "okay" without getting to get some input is the goal.