RJ11 is a 6 pin connector, using 4 of the 6 connections. RJ45 is a 8 pin connector.
So if the remote is a six pin connector it would go to the RJ11.
I have no more information than you about the unit, but it seams like you have tried all the combinations. When you say "on" what is on? You have two units a charger and a inverter . If you are plugged in I would not expect the remote switch to work, I suspect it is just for the inverter section.
2004 E350 EB V10 E/PH 2WD