Yes normally a shutdown is due to low battery charge. If you just got your tank please make sure to charge it for the recommended 4-5 hours as the wall charger is a small one @400mAh and will take a hair over 4 hours to charge from a fully depleted state.
Have you checked the voltage with a multimeter? It may be that your battery is not fully charged or being charged. When you said you tried other batteries, what types did you use? Did they have a full charge as well? Firing the cannon is usually what shuts them down as you have a combination of the airsoft unit running, track recoil, and sound being activated at the same time. Does the tank run fine for those 5 minutes and shut off when you fire or does it just run for 5 minutes and shut down?
Unfortunately without a multimeter it is a bit hard to diagnose Hopefully you have one sitting around! If not, check your local Harbor Freight's weekend sales, they usually give them away free or with a small $5 purchase. I visit my local HF almost weekly!