Let me start out with these new rigs have computer everything and a module for everything else. Things may not be directly compatible with all of the cyber stuff.
Like factory on my 1977 CJ-5 and the way I wired my 1946 Willy's similar, There is a built in circuit breaker in the headlight switch. Common to both is the floor dimmer switch. If the connections on the dimmer or headlight switch get loose or corroded by water over time, I get the blinking with the headlights at the worst of times.
I feel your situation is not similar with connections, unless you severely undersized wires and connections got real hot.
I have access to your Jeep's full wiring diagrams, so I may can track stuff down.
Do you have the wire diagram to your new set up or can you tell me the model and company?
Better still, can you link me to its PDF?
This might be our starting point to find the issue.
If nothing else, we may be able to use a relay which would give direct battery power to the headlights and the relay be turned off and on by factory headlight switch.
I have and can make diagrams that might aid you too.
Do you really love your Jeep?
Tuesday, August 8th, 2017 AT 4:05 PM