With my version, you get as powerful an ability as you pay for in character creation. With your version, in one scene the ability might not work at all if you're out of FP, in another it might reduce rolls by a bit for a single FP, and in another it might make the most powerful supernaturals into effectively pure mortals because you got a lot of FP to spend on it. Worse than the fluctuating power is that the ability does not scale. It will be exactly as powerful in a feet in the water game as it will be for a 20-refresh game. And those two problems together make for bad ability design IMHO. Other issues below;

How would it interact with range? For example, both wizards and superstrong enemies could pick up a car and throw it at a house and not be affected by a threshold normally because they are attacking at range and indirectly. Ditto, a superfast enemy standing outside a building wouldn't gain any penalties to defense against gunners from inside the building because they'd not be standing within the house's threshold.

Does it follow the standard threshold rules? A supernatural entering a threshold doesn't actually take any penalties to rolls or whatnot. It has to set aside an amount of powers costing in refresh equal to the threshold. I.e. a 2-shift threshold means a vampire would either have to reduce their supernatural strength to inhuman, or reduce their supernatural toughness to inhuman, or reduce their inhuman speed to no benefit. The exception being wizards, who take penalties to power and control instead.

Do note that the supernatural being picks which power to leave behind when entering a threshold. They could well set aside the least useful ability for their given situation and retain those that would benefit them most.

Last but not least, a "penalty to rolls" mechanic causes issues. Strength powers and things like Claws or Breath Weapon don't give any bonus to rolls at all (weapon rating isn't rolled). Speed powers do. Toughness powers don't. Senses don't. Shapechanging doesn't - not directly. Living Dead doesn't. Flying doesn't. And so on and so forth - not only various powers exist that don't give roll bonuses at all but those that do give bonuses don't give them at the same rate. For example, Speed powers give +1 athletics per 2 refresh, True Aim gives +1 weapons per 1 refresh, Cloak of Shadows gives +2 Stealth per 1 refresh.
So if you use a "penalty to rolls" mechanic, not all powers will be as affected by the ability. (hence why normal Thresholds subtract a set amount of refresh instead)

It might work, but I'd be cautious. You'd essentially multiply the use of a Fate point by the number of scenes in a session, which might be a bit much. What's more, this way it becomes an "I have to spend this in this scene or it was wasted" sort of resource, which can lead to an overblown use of the power. With uses per session, you can spend them when you need them, without as much of a need to do so.

I think you should determine when you take the power if this is going to be a +use or a +skill thing, like when you take refinement as either focus item slots or specializations. You can switch at a milestone, of course, but switching whenever you want to seems to much. Just wanted to clarify.

And -2 refresh for +1 is kind of expensive as well.

I don't think you'll need the -1 prerequisite. Using Convictions for Declarations like that is pretty much exactly what Guide my Hand is supposed to do, so you'll kind of remove the need for spending Fate points altogether, except for very few cases. Seems weird. I would just make "Faith moves Mountains" and "Hope springs Eternal" separate powers.