My car wont stay running unless i keep a battery charger on it this is a race car so the wiring is different

Charger light to a phone will stay on because some cigarette lighters do not require a key on for use. It is wired into a 20 amp fuse though. It is best to unplug devices from cigarette lighters to prevent battery drain unless it is an everyday driver. A phone charcger does not draw to many amps for major battery drain.

batteries will self discharge over a period of time especially in cold weather
however 3 batteries replaced in 6 months raises issues
first question is why would they not charge up on a charger or a run in the vehicle
second question is the battery of sufficient size to run the electronics
have you had the alternator bench tested to check for correct out put, regulator operation and rectifier problems
check drive belt condition and tension
the wire could be a ground wire but I would think that if it is significant then something would not be working correctly

Either the battery is a dud, (cant retain a charge) or the alternator is not charging possibly due to a wire not connecting or wires incorrectly connected or both. Check voltage at the alternator (large terminal) without motor running, should be around 12 to 13 volts

Have the battery charged and tested and then test the charging system. You should never run a car with a very dead battery. that will burn out the alternator.

Check temp sensor and TPS, throttle positioning sensor.The power control module,"pcm.is a possible thought.Just check all connections. It is always a good idea to spray WD 40 or equivalent in the connections.

I had the same problem. My car sat for a few days and when i went to start it i heard an extremely loud pop (kind of like a backfire). After that it would start and sound fine, but die 2 seconds later. Thank god my bf is a mechanic he knew instantly that the pcv valve hose that connects to the intake manifold unplugged itself. He plugged it back in and it runs fine now. I didnt however get a check engine light, but i only tried to start it twice before he fixed it because i was scared it jumped time.

Sounds like a faulty alternator, by increasing the revs,you are increasing the output enough to keep it running.

sounds like you have a fule pump fault,,,the battery fault is strange??
unless the pump has a short on it, draining the battery,,,
the alternator will very soon pack up if you keep running it without the battery hooked up

try a compression test on the cylnders, If it overheated when the head blew , the rings can go soft with too much heat

Will not idle unless gas is applied?
Will stay idle after a bit of running?

First i'd recomend starting off with a new battery if yours is older, durangos have a quirk with voltages and sometimes wont play with 12v.
Also the TPS could be an issue.