Sorry for the delay in responding, but health issues come first. My first reaction to your voltage measurements is how it could be that while on shore power only; converter is running and charging, since both battery strings are above base (full charge) level of 12.6vdc for resting wet cell batteries. So 13.1 vdc on coach and 13.6 on chassis? The way this BCC works is when on converter the coach battery charges to approx 13.2 vdc at which point (after 15 sec delay) the BCC voltage sensing electronics closes the isolator relay, which connects both coach and chassis batteries together in parallel essentially. At that point the charging voltage from the converter should be the same value on both strings, if all is wired correctly and functioning normally.
So charging current is coming from the converter through the 50 amp re-settable circuit breaker in the BCC and feeds to the coach battery through the "on" coach battery disconnect latching relay.......and also supplies charging current to the chassis battery via the now closed isolator relay. I modified a charging circuit to show roughly how the RV Custom BCC connects your battery strings (see attached).
When you operate and release coach and chassis disconnect switches at the door you should hear the latching relays also operate and release. Similarly when you operate the emergency start switch by the driver, you should hear the isolator relay also operate and release. If there is no sounds from any of these operations then something has failed inside the BCC or something is wired wrong.
Strange also that DC lights work when on shore/converter, but when on coach battery only they fail after 15 seconds and your coach battery voltage drops under load (normal) but the chassis battery jumps to 14.2. Impossible. The shore is off/ converter is off how can the coach batteries be charging? Also when the coach lights shut off what sounds do you hear......does it sound like the coach latching relay dropping out? I suspect you have some wiring problems minimum either to the BCC or inside the BCC. When on shore/converter power the voltage you measure at the converter fuse panel should be almost or identical to what you measure on the coach batteries. Enough said for now.