Personally, I think it should only be a temporary thing, and not a permanent thing. You can fight through wear and tear, but you can't run from it forever, eventually it catches up to you and you have to pay the price for pushing yourself past your limits.
So if you decide to allow a stunt like that, it should either:
1. Come with debt (1 debt per 2 shifts of stress, and weariness related compels to call in the debt); or
2. It should hold off the stress and then apply it again at the end of the scene. If you go with this option, then it will probably end up applying consequences, which I find the more thematically appropriate of the two.
As far as your options, I think the once per session, and spend a FP to do it, works well. However, if you make it a temporary thing, like I've suggested, then I think just once per session will be fine, no FP needed.