Not sure about the dog- that's weird.

As for not giving the script, you have to see if the fit is proper before giving a script. If the diameter or base curve is not a good fit you can get some significant problems from CL wear- including corneal ulcers which can result in severe vision loss. No good optometrist or ophthalmologist is going to give an rx without verifying the fit first in a new contact lens wearer.

I have seen people get cosmetic contact lenses from beauty supply stores (who should never sell them) and come in with severe issues on many occassions because of poor fit. It can be bad news. the problem isn't ever the lens itself, it is someone is trying something on their eye they have no business wearing without knowing it is safe.

As for the tech becoming frustrated within a few minutes, that's bad form. It can take some kids over an hour to figure things out the first time, and it should be expected that a new contact lens fitting will take a long time. Sorry your daughter had a bad experience!