Edit: I am not advocating "don't use Fate" here, but rather, see if there's anything from House of the Blooded that can be implemented as house rules for more competitive players.
You may want to check out House of the Blooded (HoB)- it is not 100% strictly Fate, but it does have Aspects and all its various usage - invoke, tag and compel, and it is designed with antagonistic play as an option. You may be able to take some ideas from how Aspects is implemented there and add it to your game.
Aspects are Strictly Defined
Each Aspect has a defined effect when tagged, invoked or compelled. For instance, if you have the Aspect "Seeker of Secrets", you have to define what it does when it is compel. Your opponent will always cause that effect to happen when compel or tagged.
The author of HoB mentioned the reason for this:
My experience with player-vs-player roleplaying games -- and I've
played in a lot of them -- made me realize that Aspects are were too
open-ended for that kind of environment. In a PvP game, every player is looking for any advantage, looking to exploit any weakness,
exploiting every loophole, interpreting vague or unclear rules to far
extreme to protect their own characters and crush those who they
...snipped... We need hard and fast rules or those tense moments when
characters face off for the last time. And so I took those multi-dimensional Aspects and turned them down a notch.
That said, the author did say if the GM is prepared for it, Aspects can be used as per in Fate.
Aspects are kept secret.
Aspects, in both HoB and Fate, are powerful tools that can be exploited. For a competitive game, that is even more so. A player may gain the 'right' to use an Aspect against you if you use it in play and it is observed by his character. He then needs to spend a Fate Point, and make a contested roll for his character to learn the Aspect.
Note that this is a roll for "in-character knowledge". You may see the Aspects written on your competitor's sheet, but you can't use it until your character goes through the process. As in Fate, players tend to know each other Aspects, there is a need to separate player's knowledge from character's knowledge, in a more PvP/competitive setting.
Winner of the roll narrates what happens.
There are also additional rules for how much you can narrate - usually by taking on additional risk. In a contested rolls between players, the winning player has the privilege of narration. If the losing player took additional risk before the roll, he can narrate one detail.
This allows players to narrate the outcome in a way that is desirable to them (and whatever nefarious plots they have for other players), without having to letting the GM knows their plan. Players may narrate in a way such that they can force their competitors into a more vulnerable position (due to their Aspects).