IMO, LiFePO4 8s "24v" is the best, as that enables the EB3A to charge at roughly 200W thru the DC input port. You should be able to simultaneously charge the battery via solar with an MPPT controller. You may need to limit charging voltage of the battery to the EB3A max of 28v vs. common value of 28.8v. However, Hobotech on Youtube showed his EB3A ok at 29v in.
I tried charging mine with a 12v battery and it works ok at high states of input battery charge, but once the voltage drops toward 12.0v it loses the ability to charge. As I recall it actually seemed to waste it's own battery trying to charge. So you could not get the full capacity from a 4S LiFePO4 battery. Several Youtubers, like Jasonoid, have demonstrates how adding a 12v->24v converter between the 12v battery and the power station solves that issue and allows for much higher charge rate. However you do pay a conversion penalty doing it that way.