I don't know about asthma medically so I don't actually have advice on that. Talking with your child's doctor would be a good idea before you go further in considering a pet. Your child's doctor will have better information on all of this than any of us could have.
And of course you know this but I want to encourage you to do as your heart and mind lead you to do on this matter. I am absolutely certain that, you are fully aware that it is your kid who suffers when allergies come up for her, and not you!. Nor would you ever want to do anything that makes your child suffer. That is the whole reason that you are thinking so hard about this decision, weighing the options, and have asked for the opinions and experiences of others. You don't need anyone telling you what to do, just information to help you make the right choice.
In my opinion, if you had to rehome the dog it wouldn't automatically teach your child that animals are disposable. In my family as I grew up there was a dog rehomed -- for good reasons. He stayed the rest of his life with the new owner and benefitted greatly from that change, as did we as a family unprepared to deal with that particular dog.
It did not teach me that animals were disposable. It taught me that animals are to be taken seriously and if for some good reason it is not working out for the animal to stay in one home, another very good home can be found, and everyone can be happier as a result. It taught me that it is vital to take everything into consideration before getting an animal companion, and that the feelings of the animal must be taken into account always, and that you are responsible for giving that animal the best home possible, whether it is with you or with another.
As an adult I have also rehomed an animal, for a very good reason, and again everyone was happier. It's not something I take lightly, but equally I do not believe that to take on an animal means to keep that animal for life, no matter what, if the situation is not appropriate for either animal or person. sometimes on this forum we recommend to people that they re-home their dog. Rehoming an animal doesn't by any means necessarily involve anything to do with rescue, as you can find another very good home yourself, and that is what I did when it was necessary to do so. Naturally one would never abandon an animal or take one to the shelter, which is mostly where rescue enters the picture.
I would never be one to say, Oh just get it and give it away if it doesn't work out! As I say, I do not take it lightly. But in this case, it seems worth taking that risk. But that is only because you have said that she can be around yorkies for days with no reaction, and it is only my opinion. The opinions of those of us here are only good for offering things to think about, which is what you came here for. You alone will know what the right decision is.