some definitions so everyone is on the same page:
chassis battery should only run the vehicle engine.
coach battery should run the camper stuff and the generator.
There should be a solenoid or something that connects all the batteries when the vehicle is running so that the alternator will charge everything, but when the vehicle is off the coach and the chassis batteries should not be connected together; your generator only should have the coach battery available.
When you say, chassis battery, are you talking about the battery dedicated for the camper (you have the terms backwards)? The battery for the vehicle should not be a factor at all for the generator unless you are pressing the "jump start" button to try to get more amps. Many RVs do not have this feature though.
12.45 is not a healthy battery, but is still fine to start most vehicles. Charged up should be around 12.6 when it was new. Below 12 is pretty low. 11.6 is considered dead.
12.4 should be plenty to make the generator happy for starting, but have you measured how low it drops when starting? Sounds like it starts ,so the battery should not be an issue. The generator will make 12 volts DC (or 13 or even 14) so that will be what the generator sees despite a lower battery.
I suspect it is not a battery issue.