Since you are building your system with inspiration from FATE, let's look at how this could be done in FATE as it is written. I'll assume the Dresden Files iteration of the rules as that is what FATE Core is being based on.
(I'm also going to address the general case before dealing with the specifics in your question).
The thing to remember is that headshots are hard (and should be, see below). You need to deal enough stress to the target to blow past their stress track and force them to at least take a consequence. This requires some manoeuvring, for example…
- Go shopping for a nice sniper rifle (gain a weapon that deals +3 stress)
- Also get a good telescopic sight (manoeuvre, resources check, telescopic sight on my gun)
- Hide (manoeuvre, stealth check, he'll never see me coming)
- Aim (manoeuvre, guns check, in my sights)
- Wait for the perfect moment (manoeuvre, discipline check, standing in the open)
Then the PC fires. They get their gun skill, the roll of the dice, +3 stress from the weapon and have 4 aspects they can tag for free. Assuming they are Good with guns, that's an average of 14 stress less whatever defence the target manages to put up. Say they have a stress track 5 boxes long and net out on 2 for the defence, that's 9 past it and they have to take a moderate and a serious consequence to avoid being taken out … and at that gives two more aspects that can be tagged for free. The shooter could also spend fate points on whatever other aspects they have available to them.
Alternatively, if the GM agrees, tag the aspects for effect and narrate the one shot kill. The player earned it with all the manoeuvring in advance.
On the other hand, the PC might be facing some mooks (really not the situation you were talking about, but I'll mention it for completeness). They only have a couple of stress boxes and can't take consequences, so there is nothing wrong with narrating them being taken out as taking a bullet to the face.
Earlier I said that headshots should be hard. If they were easy, then there would be no point in being good with guns. You can just take headshots all the time. (It's the age old problem of balance, if there is a cheap but effective tool, everyone will take it at the expense of interesting alternatives).
And your specific points, and how they relate to what I just wrote…
Typically, this happens when a player feels her character has a high amount of precision but perhaps less stopping power
In a system modeled after FATE, stopping power is represented by the weapon bonus and is layered on top of precision. The trick is to make sure that the precision really is good enough, and that is what I dealt with up above. Getting a snap shot off at somebody's head is difficult, so it requires a really high result to pull off.
or when a player feels an opponent is unusually threatening and resistant to normal approaches.
A simple exchange of gunfire isn't going to the job. Taking an approach that isn't normal means manoeuvring. Get cover, get distractions, get help, get the opponent stuck in tar, and so on. Build the aspects up and then exploit them.
Allow them, but give them a disappointing result. Shooting someone in the head doesn't kill them, but just makes them skip their turn or take double damage or get a penalty to act. Tends to hurt suspension of disbelief.
A disappointing result isn't a bullet in the head that has little effect. It is a shot almost hit the target, but (because they moved) only winged them causing blood to run into their eyes, or shattered the mirror beside them causing them to turn to look at what happened and be distracted, or exploded the bag of flour next to their head and filled the air with dust.
Allow them, with dramatic effect, but at a low success rate. If headshots kill instantly, you can give them a massive to-hit penalty to keep them from becoming a dominant strategy... which makes them pointless to use and takes away their possibility for interesting play.
This is one reason I like the FATE system so much. The precise details of what happened come after the roll and the use of aspects. It doesn't give a binary result. An ideal result is that the bullet kills them. The worst case result is that it misses entirely. But there are many positions in the middle that hurt the enemy without being a complete success … and if the bullet misses, enters their shoulder, angers them and makes the PC who fired the shot a target … well, that's dramatic.