Well, actually, it is theoretically possible, but it will take you a while...

You could go from T4 on the Eagle shuttle bus to the remote terminal, transfer to the T5 shuttle, go down the escalator and across the tunnel to T6, then walk to T7.

Now, I'm not sure you can talk your way onto the Eagle shuttle bus without an Eagle boarding pass, but I'm guessing the contractors around won't put up too much of a fight...

Yesterday I did Eagle -> DL -> CO and just that took 15 minutes from the time the bus left, so I would say if the above works, give yourself at least 30-40 minutes.

To second what was said earlier, the walk back outside security isn't long - I walked from the PC in T6 to the AC in T4 including security in about 15 minutes.