Driving into work, it occurred to me there is a very simple circuit that could easily work:
- 100 A diode - inverter output to 'white wire' - this will typically drop about 1.2 V making the output voltage closer to 12.6 VDC if any current flows from the inverter
- 13.9 VDC DC-DC converter output voltage - except for any possible voltage drop, the deep-discharge battery should provide the bulk of the power. Then when it goes 'dead', the vehicle inverter picks up the load.
You'd still need some sort of indicator that the deep-discharge battery is drained. Also, you'll need a 12 V. battery to provide starting power if the deep-discharge battery is 'out of juice.' ... Just a thought.
Bob Wilson