So only a footpath for access? Not even two tyre tracks across a field?
Not an old keepers cottage is it? I'll assume it is as your description matches well.
The 1:25000 OS map does only mark a path and along the foot of crags/cliffs and still quite a steep slope down to the sea. Think one would have to see what that path is really like. Maybe they built the lighthouse/cottages by bringing stuff by sea, no obvious landing stage now. An old cart access might need a decent 4WD rather than a car to negociate it, that's assuming any access rights allow vehicular access...
Idyllic until the north atlantic storms arrive in winter. Tucked nicely away but the NE is still exposed.
Access would be the biggest snag, walking a mile with your weekly shop won't be fun. Having said that there is an inlet between the main island and the island the light is on, might be possible to use a small boat.
There doesn't appear to be a fog horn associated with the light. B-)
Wind and big battery bank. Looking at the terrain to the south solar PV might be even more hopeless than normal in winter as the south elevation may not see sunlight. Tidal flow something in the inlet? Small diesel set as back up for when the wind don't blow.
Lighting using modern LEDs and drivers designed to work directly from the battery voltage, probably 48 V.