The DC system is pretty much independent of the generator...or, more precisely, if the generator starts and runs at all, the DC system is sufficient for it.
The generator in an RV is usually not set up to charge the batteries directly, but rather the converter charges the house battery from the AC power that the generator produces. You probably aren't seeing any charge current because the transfer switch hasn't switched over to the generator, either because it's producing no output (and hence shutting itself off) or because the time delay hasn't been satisfied yet. That should take care of itself when you get the problem with the generator sorted out.
There are several reasons why the generator might shut itself down to protect itself, some of which depend on the spec level of the Onan. If it doesn't detect output it will shut down because this can indicate a short in the generator wiring that, if left running, could cause a fire (perhaps among other reasons). Some have other sensors as well for e.g. oil level.
The house batteries should charge from the vehicle engine alternator when the engine is running, it's true, but that's a completely separate system from the generator. There's some sort of isolation device that only connects them together when appropriate, either a high-current diode or a relay/solenoid based system which may have very simple controls or somewhat more complex bidirectional controls (to allow the chassis battery to charge from the converter if needed), but neither is directly related to the generator operation.