OK, so we can rule out the engine running being an issue and the headlight theory. It almost certainly looks like you've got a straight forward short to earth (most probably a frayed wire) somewhere in either the indicator circuit, or more likely the sidelight circuit which shares fuse 1. When the bike moves off the movement/vibrations cause the bare core of the wire to make contact with the frame and shorts out the circuit.
The only thing for it is to go through every wire in the sidelight and indicator circuit and visually check to see where the problem is. This will almost certainly need the tank to be removed.
I can give you some pointers to common areas of abrasion to check first:
- The front sidelight wires behind the plastic cover beneath the dash - they can get trapped/nipped up when the cover is refitted. They can also get trapped between the fairing subframe and the cowling.
- Front indicator wires from the grey plastic panels to behind the headlight - again they can get trapped/pinched when the panels are refitted.
- Indicator/tail light wires from just beside the battery inside the LH top frame rail to the rear of the bike - can get trapped by the undertray and tail piece.
- Wires/connections into the numberplate lamp - can break due to fatigue and corrode because of direct contact with road spray.
- Main loom under the tank along the RH frame rail - well known for abrasion and fraying.
- Check wires to all accessories and/or where they have been removed - be wary of any Scotchloks anywhere as they are notorious for breaking down where the insulation has been pierced and also if the fold-over flap is missing.
That'll give you something to do!