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If I'm reading this right you are trying to get a constant live feed from a hallway light circuit which is 2 way. It doesn't work because the live on the hallway circuit changes when the other light switch in the hallway is operated. In other words either the Line or the Strapper could be the live you want for the outside light but not at the same time.
In short you need to get a live from the lighting circuit to the outside light or give up 2 way switching in the hallway. Do spend 5 or so years lying in bed at night trying to work out an alternative though, it's a useful way of avoiding sleepless nights :)
Arghh, two hours trying every conceivable option and no luck. The existing switch/circuit is wired like this now, the switch I'm trying to connect to is the first one:
I can't for the life of me figure out how to connect the new switch and light to this. The closest diagram I can find is designed around a 1-way switch rather than 2-way...