Use transistors instead of gate driver for gate drive transformer
can these transistors arrangment drive the gate drive transformer
  properly


Look at this: -



If \$R_L\$ represents your transformer primary then the average voltage it receives will be half of Vcc or 6 volts. You MUST NOT allow this to happen with any transformer. To remedy this put a capacitor in series with the coil to force the average voltage on the primary to be 0 volts.