The center mount Brake light for USA models are LED types - so do you need to replace the high-mount center brake light?
Or the ones just above the bumper?
The High mount you cant' fix, you'll need to "swap out" the whole LED lamp as one unit - means you'll have to remove the upper trim holding it.
The boot carpet for rear brake taillight assembly "tucks under" a screw that holds/crimps it to the plastic trim in a 5-door model. This normally doesn't cause a problem but it can get in the way.
That screw is by (In the Hatchback) the place where the trunk cover rests to the rear of the hatch - just remove the screw if needed to help lossen the plastic trim to help loosen the carpet.
The boot carpet - if it's never been moved, might just be "stuck" with a heat set glue that keeps it in the form it's in.
The taillight connection is behind this - as you know, but if you have a TORX (Star bit) size of #25 you can go after the exterior trim screws holding the taillight lamp assembly in place which might allow for some clearance and inspection to gain access past the stuck carpet. (That is if you have the taillight "bolted" to the exterior)
The internal wing-nut type uses "grommets" to act like gaskets to keep the dust out, but if the carpet got "pinched" into one of these grommet holes then the taillight needs to be gently pried off the seal to loosen - but this can break the wing nut - so don't overdo it.
I've seen people use a simply utility knife to "cut a slit" into the carpet to obtain ACCESS to the wing-nut so they can remove it, to pull the taillight out. so they could get the taillight off the car in a junkyard, so that is an option, is to cut a slit, but I'd try to see if the carpet can be removed by prying out the hatchback - trunk seal trim that keep the water out and see if you can find the carpet ear to continue pulling off to get inside - at that wing-nut.
In case anyone wanted to know...this is what a 2015 ecoboost manual says about the process...