striker

we have the alarm running all the time, and the car isn't driven that often. so we trickle charge it. the first time my dad(it's his car) was down the street putting it through 2400rpm in 2nd, and it died. we played around with the fuses, namely the fuel pump fuse(it felt kinda loose). it started right away. so we thought well it was just the fuse was loose. today(charged it yesterday), 20 feet down the street same thing. my dad left the key in the ignition and was pushing on the fuse boxes(the one next to the battery) and the door beep came on and it fired up on one try. It only dies after charging it. never has it happened just normally starting and going. could it be a problem with electical system? do we need to ground something? the car has a dp and a m2 box. thanks in advance.