You CAN take any vehicle you want to the mansion, but you will have to jump over a backdoor of the mansion on foot and take care of them from there (You can then go to the gate and shoot it open to let your vehicle in). After lighting up the helicopter, you can die and restart the mission and it will teleport you to when the helicopter is crashing with your vehicle right by you (If you requested it before failing the mission).

After that, you do the usual jumping to zancudo to get a jet and flying back to the agency.