House battery, by definition, should not be starting an ( any ) engine. ( Unless temp jumpered... )
But the general idea, in any system with a house is for the Main engine to charge it's personal start battery. Then when the voltage reaches a certain threshold, indicating that the start battery has achieved the desired charge state, an isolator device then allows the engine to charge battery banks "down stream" of the start battery.
The "isolator device" may be either a diode device or a relay device. Each has it's strengths and weaknesses; neither is actually "better". they are simply different.
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That said, Li batteries have some great strengths, but they DO need care in how they are charged. I would seek guidance from your Li battery provider, since there are different types, with different needs...
This area is not yet generic enough to just say grab a "XXX" and plug it in. There is more to it.
