Extending the NRF24 range / power
How are my chances to modify this device to output drastically more
  power?


The device is a transceiver and the transmit/receive arbitration circuit is inside the chip and therefore you cannot put an amplifier in the antenna feed (unless you are highly skilled and have access to good equipment) because it'll stop reception of the RF signal. 

Plus, adding an amplifier (even if you could) will very likely take it beyond its legal power output rating.