From the writer, Fred Hicks:
biomancy is a tricky, poorly documented part of magic in the source
material. But, yes, it’d incorporate transformation magic, and its
healing elements likely manifest as accelerated medical care, only
occasionally branching into true “regeneration” type stuff at the
extremes.
With that in mind, I think that a Biomancer is more akin to a doctor than a cleric, which seems to fit the source material. Keep in mind, that this is an urban fantasy, so having an easy way to heal an extreme consequence (an arm being taken off, for instance) brings to mind the question of why is cancer still a problem, and (without going into too much detail) why are extreme wounds to characters in the source material such a problem.
That said, I'd look towards the character concept and the focus of the player in creating the character to how you might tailor things to make the concept work in your game, and make things satisfying to the player in combat.
In the Dresden Files combat, there are four basic conflict actions: Attack, Maneuver, Block, and Sprint. (ref YS199). Taking sprint out of the equation and reducing the actions to their simplest intent, these actions are meant to either protect the character (thus prolonging the conflict), damage the opposition (thus contributing towards the end of the conflict), or set up opportunities for one or the other.
An application of Biomancy that would fit within these rules is a Block. If someone is in combat, the Biomancer could focus himself on increasing their natural resistance to damage and their natural recovery. The description of the effect would reinforce the use of Biomancy i.e. instead of a kinetic shield stopping the bullet or altering it's path, the bullet impacts the target, but the biomantic energies were able to knit the damaged body before the effects of the shock could even register.
If this is not enough to satisfy the concept and you want to extrapolate the rules and/or use the FATE SRD rules, you can move beyond this. An example of mitigation of consequences is already given- and I think given the statement above about what the intent of what biomancy would be fits their example. So I'd concentrate on stress.
Looking at YS201, we see a definition for stress:
Stress is an abstract representation of the difficulties that threaten
to take someone out of a conflict. In a physical fight, stress can be
minor cuts, bruises, fatigue, and the like. In a social or mental
conflict, it might be loss of willpower, composure, or emotional
control...
As attacks inflict stress (which result in consequences), and blocks inhibit stress by increasing defense (resulting in less consequences), the only actual result of removing stress is to extend the conflict. This is the balancing effect to keep in mind. If one side has the ability to remove stress, and extend their staying power in the conflict, and the other one doesn't, you tip the balance in the favor of the side that can remove stress, though the fact that this is a transference in the case of non-rotes does mitigate this concern some.
With that caveat in mind, the first place one could look for an example of what could be done with removing in-combat stress is the FATE SRD. In the FATE SRD (based on Spirit of the Century), there is a skill, Science, which has a stunt Medical Attention [Science].
When using Science as first aid in the middle of a fight, the
character must take a full action with a target who’s not trying to do
anything else active (i.e., forfeiting his next action). Make a roll
against a target of Mediocre; if it succeeds with at least one shift,
the subject may remove a checkmark in his one-stress box on his
physical stress track. Every two shifts beyond the first improves this
effect by one; for example, with five shifts, a character can remove a
checkmark in his target’s three-stress box. Success can also be used
to “stabilize” someone who has taken a severe or lesser consequence
that would appear to be life-threatening (e.g., a Bleeding to Death
aspect) – in game terms, this has the effect of limiting the extent to
which the aspect can be compelled. A given person can’t be the target
of more than one first aid action in an exchange.
Under this approach, one would use an Evocation with the spell's intensity against the target of Mediocre and follow the same guidelines.
Another approach one could use is to look at stress as taken as an intensity of the blow, especially since each dot of stress is an increasing value, and the lower levels of stress won't help with higher intensity wounds. So if your Biomancer wanted to give someone the ability to clear their 4 stress box, they would require an effect with a 4 intensity. Knowing that, you can craft an evocation spell against that intensity.
Note that for balance's sake, and because this isn't explicitly laid out in the rules, the intensity might have to be altered, especially given that a moderate consequence removes 4 stress from the hit, and a mild removes 2, so the equivalency might be a bit different. But I think that these general guidelines would fit the spirit of the rules.
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
Yeah, there's options here. One of those is actually conceding; your group has misunderstood how that mechanic works.
Option 1: Get Taken Out without taking consequences.
When you have to resolve an attack, you don't have to actually absorb the stress. You just choose between absorbing all of it, or not absorbing all of it and getting taken out. You can even absorb part of the stress, and also choose to get taken out, if it seems narratively appropriate.
This is kind of a good thing because if you were actually obligated to absorb as much stress as possible, every single fight someone's in would be horrifically and permanently character-scarring via extreme consequences, and your story would be filled with characters who need magnitudes more than the usual amount of therapy and trauma counselling.
Fred Hicks himself confirms this is an option in an answer here. He also points out this is a lousy way to avoid harm, since when a character's Taken Out their opponents have complete control over their fate — death and complete and total destruction of the Taken Out character is an option on the table. That's mainly a player's issue though. In this circumstance, what's more relevant is the characters could say these NPCs are concussed out of the mind control (they're free!) and even join the heroes of the story to exact revenge on the person controlling them, or something similar. Anything's on the table.
What this looks like in practice: Decide when the fight is narratively appropriate to end, and let that last punch end it.
Option 2: Concede, because it's not an in-fiction action.
From what I'm reading, you've interpreted concession as an in-fiction action, that the character themselves is saying/gesturing their surrender to other characters, that the fictional characters are fully aware a particular character is voluntarily giving up. This isn't the case.
Like most of the other Fate mechanics, this one takes place above the game, at the table between players. A player says they want a character to concede, the player names the worst parts of the character's fate they want to avoid, and the group then works out how things shake out in the fiction to let that character get taken out of the narrative and avoid that worst stuff.
The Fate Core example of concession demonstrates an example of how this plays out:
In this narrative example, it's not the case that Landon says "Hey! Wow! Og, you're too strong for me. How about you knock me out and take my sword instead." Instead, the players are just talking, Landon's player wants Landon to concede, and they decide the way that should work out is that Og knocks out Landon and might even think he's already killed the guy. Hooray! Og, being a character in the game, is none the wiser about the mechanics that just took place that let Landon live.
This clarification by Leonard Balsera, Fate Core's primary author, further describes the player-facing nature of Conceding.
What this looks like in practice: You decide the mind-controlled NPCs have taken enough punishment and say you're conceding. You say what you'd like to avoid for them (such as: you don't want them dying, or captured, or something) and you and the other players as a group work out how this happens: the mind-controlled characters may get tied up, locked in a room, knocked out, or escape out a window (or get thrown out of one). They never gave up, but meta-game, they've conceded and aren't in the fight any longer.
Just as with being Taken Out, the option of the mind control ending is within the confines of what the players can dictate here. That's unless you chose it as a worst part of the characters' fates to avoid, but you probably won't for two reasons: (1) it's probably not the worst parts of their fate, and (2) you're conceding which means there is bite and you can't have everything your way.
I'll note that a player in control of multiple characters (usually that's the GM) doesn't have to concede for all of them at once. It's on a per-character basis. So you can have Mind Controlled Dude Number One concede, and the other keeps fighting on — and they may also concede or get taken out.
Option 3: Don't use the conflict rules for this to begin with.
Despite the name, physical conflicts don't have to be modelled by the Conflict rules. Reserve the Conflict rules for really pivotal, awesome fights that are super worth zooming in on and engrossing yourself in. If it isn't that, the Conflict is often going to be a pretty boring slog-fest, and my group's learned that one the hard way.
Instead the golden rule suggests you can just model the strife between your characters and the mind-controlled NPCs as a Challenge or just a set of opposed rolls: the goal is to find out who overpowers who and how.
The Avatar: The Last Airbender TV series is an excellent example of physical fights that shouldn't get resolved by Conflicts. Very frequently, the heroes have something they need to do (warn the Earth King about the drilling machine, reach the Fire Nation Palace & find the Fire Lord, reach Avatar Roku's shrine in time for the Solstice) and there are people who are willing to use physical force to stop the heroes from succeeding. However, the heroes are only focused on reaching their goal and using the minimum effort to subdue or evade their enemies: this is prime material for just setting up different enemies as a series of opposition steps in a Challenge, wherein successes get narrated as a whole bunch of people getting beaten up and knocked out, or subdued or evaded.
Avatar also has some really serious fights where things are deeply personal and the only stakes are someone needs to go down: Aang versus Zuko, Aang versus the Fire Lord, So Many People versus Azula, etc. These are gripping, character-defining, plot-defining moments and are worth all the attention and investment of the audience and exploring in depth. Those are the kinds of things you should look to turn into Conflicts.
Challenges and other modelling can still result in consequences or worse if it makes sense: the silver rule provides consequences as a textbook example. If something happens that seems consequence-ey as a result of a challenge or contest or opposed roll, it can be modelled as such. If I physically overcame one of those mind-controlled guys by tossing them out a window, it could make sense to give them a severe consequence (Lots Of Broken Bones) or say they're dead. The Silver Rule says if that seems fictionally appropriate, you can make that the case, and don't let the rules stop you doing that. This means during Contests, Challenges, and other types of modelling, you can still use Consequences — you just don't have to.