The two electric toilets I've had were not manually flushable. But neither of them had a straight path into the black tank, either, so even manually depressing the ball in the bottom wouldn't have been much help -- it needed the pumping action to actually get the effluent into the tank. When you say it's "hooked up to a battery and needs to be charged via 12 volt" does that mean it has its own separate battery or does that mean it's hooked to the trailer's main battery (the one that runs the lights, water pump and more). If it's the latter, then the battery will charge whenever you're plugged in to shore power. If it's the former, then it's a really weird arrangement, and the answer will depend on how things are wired.
Apparently there are several different 5th wheel models of that Redwood brand called Crossroads, ranging up to 38 feet, and there are different floor plans, so the model number might help someone to give a more specific answer.
