Does your M416 have a military or civilian 6-pin connector? Either way, you have two primary options: (1) make your own pigtail (easy to do), or (2) add the appropriate 6-pin connector to the rear of your jeep in addition to the 4-flat already there. You shouldn't have to rewire the entire trailer.
On my pickup, I have 7-pin, 6-pin and 4-flat connectors so that I can tow nearly anything. (Previous owner of the truck installed a programmable brake controller, so I can use most electric trailer brakes.) On my jeep, I have a 4-flat tucked away "just in case," 4-round and a 6-pin connector. I had originally intended to use a 4-round on my Bantam trailer, but when I won an ARB fridge in a raffle, I decided to use a 6-pin connector so that I could charge the trailer battery when underway. (Could have used a 7-pin, but didn't need the extra wire for trailer brakes and I decided that in the event of trailer theft it would be less likely that the thieve's tow rig would have a 6-pin connector and I didn't want to make it easy for the thief to have lights.)
My two cents.
