The most common failure in that installation is the ground wire. I don't expect that sportsmobile added the remote switch with the unit, or you could try some things or look for error blink codes.
So check the ground wire from the ACR 7620. It may be fused, I doubt it, they are notorious for making bad ground connections. After that all the other wires can be disconnected. There are no sense wires, only a control wire. Sportsmobile has notoriously been known to wire this incorrectly. They do try to wire it to the "Start" circuit, and sometimes wire it instead to the "Run" circuit. The Start Circuit has 12 V when the key is in start, the Run Circuit has 12 V when the key is in run. The yellow wire will be for LED indicators , the brn, org, and green are all for start isolation, if that was wired in the brn wire would go to the start circuit.
The red control wire works like this, if left open attached to nothing the ACR will be in auto mode. If the wire is tied to 12 volts, the ACR will close , meaning you should be able to charge from the alternator. If the read wire is tied to ground the ACR will open, no charging from alternator.
As others have said make sure that cable from the start battery is above 13 volts ( for 90 sec) or 13.5 volts for 30 secs. Also it is important that the house battery is not below 9.6 V or you may have under-voltage lock out.
2004 E350 EB V10 E/PH 2WD