It's (relatively) easy getting the private coach you want-- you identify her/him, you work out the schedule, you pay the fee, done.
But class coaches are another problem. What if you want a particular class coach, or DON'T want a particular class coach? Are there options?
Don't sign up until you know the schedule
If you're in a program that keeps registration open, then just wait until they post the schedule so you know who is teaching what. If your rink doesn't post the coaches' schedule, just ask the coach you want. They will know a couple of weeks before the session starts.
There are three ways you can ask: A "will you please assign Coach Wonderful to X level class at [time] or, B "who teaches X level class at [time] or C "when does Coach Wonderful teach X class" (or what classes does Coach Wonderful teach). Good luck with A.
There are all sorts of reasons that Coach Wonderful can't be assigned to Class X. They might not have that time available, or may want, or be needed on, a different level at the same time. They may not have the seniority to get assigned a class if someone else wants it. The Skating Director may be unreceptive to customer requests of this nature (even multiple customer requests, and especially if the SD doesn't like the coach in question or the parent doing the asking). This is especially common with the obverse-- "please stop assigning Coach Arschloch to Class X, everyone hates him, that's why no one ever signs up for that class."
But you can always ask.
What's the cost vs. benefit. Is it better to be in a large class with a beloved coach, or a small class with the coach you don't know?
Don't move up
If you really love a coach and you/your skater moves up but the coach doesn't, just stay in the class. There is almost no skater who can't benefit from another round at any level, and if they love the coach that much, they might be receptive to this.
Get over it
Your child is going to have to deal with a lot of people in his/her life that they don't click with, or that they in fact don't like, and certainly that they don't know. They're going to have to deal with people in authority who don't like/get them. Shyness/discomfort with new people is not an excuse, because it's completely unrealistic to control your child's contacts to that extent. So to some extent this is not just a skating, but also a parenting issue. Unless it's a question of abuse, inappropriate behavior, or actual shirking of responsibility on the teacher's part, a skater (child or adult) who refuses to get something out of a class because they don't like the teacher is not a disatisfied consumer, but rather a spoiled brat.
It's wonderful to have that teacher whom you love, but not always possible. Go ahead and ask, but then just take the class with the new teacher. You might discover that they are wonderful, too.