So you have proven that the alternator is working and is charging the chassis battery OK. You seem to have proven that the interconnect (or emergency start) relay is operating (makes noise at least) when the emergency start button is operated by the DW…….but that relay operating is not connecting the chassis battery to the coach battery as it should. So I would suggest finding the interconnect (emergency start) relay and verify the contacts are good on it (might be burned or damaged) and also that your power cable connections are good on both sides of the relay as a start.
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