Got lost with the second solenoid. But still want to know about why the loss of voltage as was asked about earlier. Seems to be a separate "issue". (loose belt eg)
You could by-pass the second solenoid and be like we are if you wanted IMO.
Our 1991 E350 in sig has only the one relay, which charges the house batts whenever the engine is on, but is open when the engine is not on. (Don't know about when the ignition key is turned but not to engine on) No extra button in our rig to push that engages house to engine batts like you have.
So in our E350, if the engine battery were dead, I would have to run jumper cables from house to engine batts (a long way, but I have some that long).
