Can I change the output power of a BT800/BT820?

Yes. First, you'll need to have access to the pskeys. On Windows, you will need to obtain the Bluesuite from CSR. After opening Pstools, you can search the pskey "PSKEY_LC_MAX_TX_POWER" and modify the maximum TX power. As the module contains an eeprom, it will store the setting, and you will only need to do it once.