corporate-wage-slave

There's a general principle here that people should be in their departing terminal, though a few restricted exceptions exist as noted above. One consequence is that once in T3 departures there is no obvious route to T4 departures (there is a route from T3 arrivals to T3 departures however, just that 99.99% of T4 passengers aren't going to be using it). The upshot is that BA will not give you admission (you're in the wrong place), CX won't give you admission (they are closed for a few months) and AA will give admission (I suspect) only on the basis of any Admirals membership you may hold, not oneworld, frequent flyer or flight status.