Combine approaches and extras to create different narrative justifications for powered and unpowered actions.
Because of the limitations of approaches this probably won't be really viable for a long-form campaign story lasting months-worth of sessions. Still, if you're going to be playing shorter games (a month per campaign, tops) this is an elegant option. If you want to use skills, read this anyway because I'm going to bring skills into it at the end.
I've experimented with this for a werewolf game, actually: by using approaches, your characters can have the same problem-solving attitudes regardless of their form: a Forceful cowboy turns into a Forceful werewolf, and a Clever schoolgirl becomes a Clever magical warrior. When they change form, the narrative shifts to give them justification for using their approaches in more magical and combat-oriented ways: a cowboy can't bite people but a werewolf can, and a schoolgirl can't create magical illusions but a magical warrior can.
(This doesn't work with skills because skills represent what-things-you-can-do-ness while approaches represent how-you-do-things-ness.)
You can represent this shift narrative by clever use of aspects and extras. Aspects should generally be phrased so they're relevant in both forms, which can be difficult but gets easier with practice. It's nigh impossible to give generic advice for specific aspect creation needs like this; the best I can say is that focusing on personality and relationships makes it easier to keep aspects relevant between forms.
Now, extras! There are many ways to craft extras around this. (Extras are for when you want to give effects that stunts can't handle, either because it's too many stunt's-worth of effects, or because a single effect is too complicated or powerful for a stunt to handle gracefully.)
Extra: Moon Infusion
Permission: An aspect indicating your magical nature.
Cost: At the start of each session I'm in, the GM's pool of NPC Fate points increases by 1 for each action I can use magically.
Benefit: Because I am secretly a champion of the Moon goddess, once per session I can reveal my warrior form. When I do, I gain the aspect Infused with the Light of the Moon (with one free invoke) for the rest of the scene.
This gives me magical context for using my actions (like flying, and shooting rays of cleansing light). When you take this extra, pick the actions (Overcome, Create Advantage, Attack, Defend) I can use magically.
I chose to make the extra's cost a "make the NPCs stronger" effect (increasing NPC Fate points) rather than a "make the PC weaker" effect (reducing PC Refresh) because, frankly, it's more interesting to face stronger opponents than to have your own power balanced out. In play it's effectively similar: the opposition scales according to the power the PCs bring to the scenario. The exact cost may need a little tweaking depending on your game. I've borrowed the basic concept from the atomic-robo RPG, which is a wealth of resources for this sort of thing.
Don't worry about including "running out of power" type mechanics in these extras: that's what consequences are for. Just as a gunfighter might take a mild consequence of All out of ammo, a magical girl might take Overcome by doubt or Cut off from my power.
But what about skills?
I started with approaches instead of skills because they're easier and more obvious to use with transformations.
On the face of things it's still pretty straightforward: Just as having a loaded gun lets you use the Shoot skill, having sparklemagic attacks lets you use it too. However, a schoolgirl probably doesn't have a lot of ranks in Shoot, so we need a new level of complexity in representing the transformation if your game uses skills instead of approaches.
Extra: Moon Infusion
Permission: An aspect indicating your magical nature.
Cost: At the start of each session I'm in, the GM's pool of NPC Fate points increases by 2.
Benefit: Because I am secretly a champion of the Moon goddess, once per session I can reveal my warrior form. When I do, I gain the aspect Infused with the Light of the Moon (with one free invoke) for the rest of the scene. This gives me magical context for using my actions (like flying, and shooting rays of cleansing light).
My abilities are different when I'm a warrior: when you first choose this extra, shuffle my skill ranks into a different configuration representing the talents of my magical form (the new configuration must still follow all the game's rules about skill ranks and caps). Whenever I reveal my warrior form, my skills change to their new configuration. They return to normal when I do.
Now we've got a girl whose abilities radically change but her aspects stay the same--so she's still the same person, but she has a different set of competencies when she's transformed.
In short, you're over-thinking it.
If I personally were to have to model this power, I'd make a new skill (dependent on the stunt as you've indicated) and call it a day. But since you don't want to do that, then yes, I feel that Will is your best bet. I don't feel that using it for a stunt is overpowered, because a lot of stunts rely on allowing you to use skills in place of other skills, and what you've described thus far is almost exactly like Zird's example Lore-based magic in the Core rulebook.
If you really need to limit the usage of the power, though (which, thematically isn't a bad idea) I wouldn't use Fate Points; it doesn't make sense, narratively, that you can only cast a spell when dramatically appropriate. I'd add a third stress track for magic, or simply have usage of the power cause Mental stress on a failed activation roll (for which I would still probably use Will).
The thing about skills in Fate is that you can pretty much use any skill to bullish!t your way through any situation as long as it makes narrative sense; an Advantage created with Stealth is no different, mechanically, than an Advantage created with Fight, and if the Stealth character has the Backstab stunt then he's using Stealth for combat anyway. Skills are broadly applicable, in other words, and Stunts allow them to be even more so. A clever player can figure out ways to use their good skills pretty much anywhere.
So yeah, in short; make it a skill, use Will, and if you need to limit it then cause mental stress either unconditionally or on a failed roll. It really shouldn't be a problem and going beyond that is probably just hamstringing the player.
Best Answer
Use the Scale rules.
There are two ways in which you can choose to apply the Scale rules—static or ad hoc. Both are valid uses of the Scale rules and each has their strong points.
Static—this means that NPCs, creatures, gods, etc. all have a static Scale ranking that doesn't change. The PCs Scale would change as you play, which would represent them growing in skill, e.g. "leveling up".
Scale ranks could be assigned broadly by groups (see below):
You could also create larger gaps between groups:
Ad hoc—this means that the Scale level of an opponent(s) depends on the circumstances, such as have the PCs encountered this foe before, how skilled are the PCs, and what the situation is for the encounter.
Here are some thoughts on that:
Scale can be used to reflect the skill level of the PCs but there isn't a 1:1 ratio between Scale and levels. I would liken Scale to how D&D 4e had paths of character advancement with 1-10 being the start, 11-20 being paragon, and 21-30+ being epic. You could think of them as being Scale 1, 2, and 3.