Is the antenna properly resonant and impedance matched to the IC? That's the first part of getting it working at longer range, before increasing power.
Edit;
after looking at the device datasheet, the TX outputs have full supply voltage swing.
With MOSFETS, look at any smallish logic level (very low threshold) type devices that can handle 500mA to 1A; don't go for large ones as the gate capacitance is very high with power types.
Try a P channel + N channel complementary pair with gates linked as input and sources as output, with drains to power and ground.