At some point you have to decide to either trust the judgement of your current instructor or find another instructor. And certainly no harm can come to you from staying with the safe and staid school horse type, but I'm a little sceptical about your instructor's motivation. She may want to keep a capable, reliable twice a week lesson student whether she has the right horse for you to teach you on or not.
Here's my reasoning behind this - an appropriate horse for you to ride at this stage of your riding is a horse that can be put on the bit when you're 80% correct in your aids; not a horse that's difficult to put together or unfamiliar with the concept. You want a horse that will reward YOU for asking correctly by going correctly; that's the best way to learn. A horse that's evasive or crammed into a false frame will be miserable to learn on; an uneducated one, nearly impossible. If your instructor can get on this school horse, and ride him on the aids after a brief warm up and keep him moving actively and correctly, then the horse is probably appropriate choice for you. If she can also explain *how* she's doing it, and that how includes skills you're working on in your lessons, then it's a good choice.
If the horse doesn't really know how to go on the aids, or isn't fit enough, or ridden correctly often enough to go on the aids and stay there consistently for a better rider, then your instructor is sacrificing your development as a rider for the sake of a steady, paying student.
If that's the case, I wouldn't necessarily break the relationship with the instructor, but I would become a lot more assertive about finding a horse to lease that you can really progress on.
And to be fair, the type of horse you're seeking is a scarce and highly desirable commodity.
Good luck to you in your riding journey, and please post back on your progress!