Ante,
There is no need to have the inverter on when plugged into shore power as the shore power supplies the AC that the inverter provides when not on shore power. So you can turn off the inverter to decrease the battery load but be sure the battery charger portion of the inverter is still on.
From your write up it sounds like the chassis batteries were low and the house batteries dead. Do you have a dash display (e.g Aladdin) that shows house battery voltage? Did you happen to notice what voltage was displayed prior to replacing the batteries?
The schematics in the 2008 Patriot wiring diagrams show that the chassis battery switch disconnects the battery(s) from the rest of the coach.. However, since it is connected to the battery with a 4 AWG cable, I am pretty confident there is at least 1 smaller AWG wire connected to the + terminal that doesn't go through the cut-off switch to keep the ECU powered. Over a month or so this small current draw can pull the batteries down. So if stored without shore power it is wise to run the generator or engine for 30 min or so every few weeks to keep the batteries charged. This is especially important if stored inside where the solar charger will be degraded or non-functional. (The preceding assumes wet cell batteries. I'm not familiar with Lithium charge/discharge characteristics).
The house battery disconnect switch does not isolate the batteries completely. The wiring diagrams show that the cable from the battery runs to a rear run box terminal and then back to the disconnect switch from a separate terminal. It's not clear to me what current draws are present there. A DC current meter may help determine if there is a draw present.
The battery switch diagrams are on pgs 155 & 158 and the rear run box is shown on pgs 33-136 of the 2008 Patriot wiring diagrams in Coach Assist.
Steve