You need to have your battery cut off switch on for the house batteries to charge while plugged into shore power. Shown lower left corner. Your battery cut-off picture is on page 7-3 of owners guide.
While your traveling the alternator may charge your house batteries.
This will happen if you have the BIRD System.
I thought for sure there was at least one 12V on my 98 & 99 KS coach's but I could be wrong. The outlet would feed off your chassis battery.
My battery disconnect will read the voltage of the house batteries on led display.
Have you checked your house batteries to see what they are reading without engine running or hooked up to shore line?
Where is your inverter located and do know what type or model it is?
Here is a
thread on 1997 coach and troubles with batteries. Never heard back from OP if he fix his problem.