Is it possible to have a charging unit and output unit connected to a battery at the same time?
It is very common to have a charging source and a load connected to a battery at the same time, so that it looks like you are charging and discharging the battery simultaneously.

What actually happens is that, if the charger can supply more current than the load demands, most current will go directly from charger to load, with any excess current going to charge the battery.

If the load demands more current than the charger can supply, the battery will supply the additional currrent the load requires.  If this condition persists, the battery will become discharged.