The American Academy of Feline Practitioners has published guidelines for the management of hyperthyroidism in cats. It’s an excellent resource and you can access it here: https://catvets.com/guidelines/practice-guidelines/feline-hyperthyroidism
I’ve had two hyperthyroid cats, and both were treated with radioactive iodine. Both were cured of their hyperthyroidism. I tried methimazole on one cat but she couldn’t tolerate it, and that’s why I went with the radioactive iodine. This is expensive up front, but long term it can be cheaper than methimazole because you don’t have to keep paying for vet visits, medicine, and lab tests. The really nice thing about radioactive iodine is that it cures the disease rather than just masking it. No more pills, no special diet, no more sick cat.
Simkie has some good comments–treat the nausea and see if that helps and maybe try the transdermal gel. You can wear gloves when you apply it to keep from absorbing it through your skin. I think the new guidelines no longer recommend a methimazole trial before going to radioactive iodine.
Let us know how your cat is doing. I feel for the little girl.