Stock there is flat wiring loom that includes the center brake light and the defrost/defogger element and rear hatch interior lighting. You might look to see if you have any spares in this loom - on the car side it runs down the passenger (LHD) B-Pillar.
The center brake light is connected in parallel to the right brake light.
The loom is on a car with center brake light installed - I assume all cars (of the right vintage) have the same loom and the wires for the brake light are simply not connected. Its quite likely the connection to the right side brake lamp is already made - but it may not be...
It is an 8 pin (T11) connector & 8 wire loom:
1 Rear Screen Defrost Heater Element
2 Rear Screen Defrost Heater Element
3 Ground (stop light & interior lights)
4 Interior Lights Fused Power
5 Interior Lights Switched Ground
6 Rear Screen Defrost Heater Element
7 Brake light Positive Feed
So you can connect the ground the brown wire on the interior light switch - but you need to connect to pin 7 of the connector for the stop light positive feed.
I'm not sure it is completely plug & play - you may find you get stop light failure errors when its connected - I'm just not sure if there was a different config of bulb monitor for this usage case.
Adding a ballast resistor in parallel to the LED's also would work with a bit less digging into the car. The LED's are very reliable and this would still warn you of some connection issues and all other bulbs. Not any more efficient than a Bulb solution but if its just for look/improved visibility that probably isn't critical. I have this with an LED bulb in the stock 3rd brake light housing.