Does she need the better boots/blades if she isn't spending tons of time on the ice? For example if she has started working on her axel but only spends four or five hours a week on the ice is the investment warranted?
I'm not the right person to answer this, since I can't do an axel. But since no one else has:
I think it's wonderful that your daughter is dedicated enough, and athletic enough to be working on axels, skating a few hours/week, in just two years! Congrats. I'm so jealous.
Fewer hours on the ice may just mean her internal ankle support isn't as good, because she doesn't get as much exercise, so she needs the support of a somewhat stiffer boot. AFAICT, it is very easy to under-rotate an axel, which could injure an ankle. But these are matters to take up with your coach and/or an expert skate fitter, because it depends a lot on her.
An axel is not considered a beginner/recreational move, and a beginner boot isn't designed to support it. Beginner boots make her risk an injury, or may not give her the confidence to jump high. But if she is very light and short, maybe your coach thinks she doesn't generate enough force to worry about... There is a such thing as too much stiffness, and too much weight, so you should probably trust her coach.
With a beginner level blade, she has to point her foot more to reach the toe pick sufficiently. She might be flexible enough to do it, though toe jump require even more toe point, but it is possible (I'm not an expert on jumps) it also requires more strength to do jumps on beginner blades, because you have to make the point more quickly, and because the center of the blade is flatter (longer rocker) on a beginner blade, then is more curved (shorter rocker) near the toe pick. A really good skate sharpener could compensate for those things, by changing the blade shape, if you haven't worn out the blades much, but few would.
What it boils down to is that beginner blades make your daughter work harder to jump. But ask your coach.
Beginner blades need to be sharpened more often, because the steel is less durable. So sharpening may cost you more (as will replacing blades, if you have to), and if you have to drive a long distance to get it done well, you need to take that cost, in time and money, into account as well.
Skating equipment is expensive. But the right tools will save her a lot of hours of skating and lesson time to reach a given skill level, and may significantly reduce her injury probability. It's also possible you are spending much more on lessons and ice and driving than on equipment, so scrimping too much on equipment can be a false economy in several ways.
If you need to economize, get her to take good care of her equipment, if she isn't already. E.g., use blade guards walking from where she puts on her skates to the ice, and back again to where she takes them off, so you don't have to sharpen and replace blades as often. Carry and store the skates out in the open, rather than in a closed bag or trunk, so they don't rot, and last longer. Don't store them in a hot car. Use a good fitter in the first place, because ill fit boots last many times less long. (Maybe she outgrows equipment before it fails - but maybe that one pair failed partly because she wasn't taking as good care as she could. Also, you can get more on the used equipment market if it is still in good shape. Besides, around age 11, her feet will probably stop growing - thank goodness.) These sorts of thing may save you more money than economizing on equipment.
(Unfortunately, she is also old enough now to realize that if she put in more hours and took more lessons, she could compete better. This is the wrong sport to take seriously if you can't afford it.)
- Just my inexpert take.