Other Fuel System Tank Assembly problems of the 2011 Buick Lacrosse

Whenever we fuel our 2011 Buick Lacrosse cxs it would not fully start and after several attempts, it would run the battery down. After the second incident, it was observed that if we rapidly pump the accelerator while the vehicle was trying to start it would pick up and fully start. This is what we have been doing and is still doing, regardless of the fuel station (exxon-mobile, shell, murphy/walmart, citgo, chevon, valero, fuel depot, ect. ). Here is a list of things we have attempted: 1) with the gas tank at about half, open the gas tank put the gas pump into the tank, don't pump any gas, close the tank, start the vehicle (maybe opening the tank was the culprit): all is fine. 2) with the tank at about half, full the tank (different station), start the vehicle: all is bad. 3) with the tank at about empty, full the tank (different station), start the vehicle: all is bad. 4) with the tank at about quarter, full to half tank (different station), start the vehicle: all is bad. My conclusion is that the vehicle is creating the problem and needs to be corrected. Before reaching this point, I had to call for road service twice. In the first case the vehicle was towed to a dealership and they found nothing and the second we jumped the vehicle.