I have a 2006 Grand CRD and tow a 2010 Jayco with factory fitted with L.E.D's. I have had no issues with lights strobing/flashing when headlights are on.
Is your tow bar and harness factory fitted?
Check the earth in the socket at the rear of your Commanderand the trailer earth as well. It is the centre white wire(3).
I ran a new earth wire direct from my battery (I think was 4mm) largest I could fit in factory socket but only because I have no second battery and run the fridge from the car.
If your earth appear fine I would then run a temporary wire from the battery negative terminal to the earth socket/plug of the trailer first and see if it has any effect.
And check the earth on the alternator as I have heard of the earth/ground becoming loose on them too.
The am fairly sure the flasher can is only used in the indicator circuit and won’t be the problem.
If you still have the issue I would get the alternator checked out, the regulator may be on the way out?
Dave