Can't see it being a battery problem because the alternator takes over from the battery when it's started and yours is starting. If it's failing to turn over and start then yes the battery is worth looking at.
Once started the ECU comes into play to keep the engine running. Since I don't understand how the ECU software is structured I'd guess, once started, the ECU begins looking at its sensor inputs to determine the best control outputs to maintain the engine.You could suspect the alternator output as well at this point since no voltage or low output means the ECU can't function correctly .
I'd also look at the ECU inputs typically, MAF for air input info and and lambda sensors to indicate if the engine is running correctly and thereby adjusting the fuel input at the injectors. I'd also clean the throttle body because my experience tells me that even small deposits affect the engines performance. Mind you I'd expect the engine management light to come on if the MAF or lambda sensors were at fault. But then again I don't know at what point the ECU turns this on if any are faulty.
If your in the AA or RAC/other, give them a call they've always been helpful to me.