Sounds like you want to look for a coach who cooperates with other coaches?
Strictness isn't everything. There are a lot of other differences. Remember that teaching style (e.g., words, demos, being physically guided through a motion, explaining the physics of how to move, explaining how to use each muscle and joint, etc.) should match the learning style of your daughter. Some coaches can teach many different ways, and are able to adapt to their students. Some can't. Also, style of motion has to match. E.g., if your daughter isn't very flexible, she can't take advantage of the types of motion a very flexible coach is accustomed to using, and vice versa. And coaches who move very differently from her current coach will be hard for her to adapt to. And there are coaches who are very competition/test oriented, and coaches who aren't, and coaches who give time priority to their best students around competition/test time.
I personally think most people should stay away from anyone who seems in the least bit abusive - e.g., who sometimes shouts at students, or other skaters and coaches. Also someone who does a lot of unnecessary touching - though sometimes touching is the only way to teach - can be very bad news.
You didn't mention her age, but your daughter may have very good ideas of what works well for her in a coach. So may her friends.
If you know any judges, and they are comfortable giving recommendations, they often have a very good idea of how well the students of each coach test and compete.
And talk to other skaters and parents. They often have very good reasons for not liking some particular coach, which may or may not apply to your daughter. But remember - most students are quite happy with their current coach, and were quite unhappy with a coach that other students are happy with. It's like asking someone whether they love the car they just bought. The answer is almost always yes.
It's a shame your DD doesn't have a lot of time to switch. Watch lessons taught by other coaches. If you like what you see, try lessons with a few of them. I wouldn't trust any coach who objected to that. Watch the trial lessons, and talk to your daughter afterwords, to try to figure out who has been effective at teaching her.
View it as an opportunity to figure out what is best for her, without having to worry about upsetting your current coach. Maybe it will be a good thing. A coach who can't work with other coaches maybe wan't very good at working with his students either?
Don't get too upset. If you do, your kid will too. It's not the end of the world.