Can I avoid Fair Access Policy by implementing a private TTN network?

Exactly. LoRa is probably not what you need. Implementing your own backend (or forking TTN) might remove the limitations for you, but will only increase data loss for everyone who uses the frequencies (because physics).

There might be a solution, which is the fact that LoRaWAN can also use FSK modulation. This will have a significantly shorter range, and we will probably not limit the airtime for this modulation.