[frodo.rok]

I think the problem will pesists if you continues to have a current transformer as your current sensing device.
Because current transformers are made to have low current in load side or sensing side.And you need a relatively high current on sensing side.
Also building of enough magnetic flux in the circuit will take time which you cannot allow.

You can solve both of this problem if you chose a PERMANENT MAGNET type relay circuit to switch on the LED circuit which will be supplied power through battery.
Just try to remember the famous experiment where 'a permanent magnet placed parallel to a current carrying coil moves clockwise or anticlockwise depending upon the direction of current', and you will get the idea.