Here the main HVAC_status object that containts all properties (even those I not understood...)

struct HVAC_status {
    word power;        //power (on, off)
    word op_mode;        //mode (auto, hot, dry, cool, fan)
    word fan_speed;        //fan speed (auto, high, medium, low, silent)
    word temperature;    //temperature (16~32°C)    
    word param_6;        // ?
    word param_11;        // ?
    word param_12;        // ?
    word swing_mode;    //swing mode (on, off)
    word room_temp;        //room temperature
    word param_102;        // ?
    word param_201;        // ?
//    word param_300;        // ?
    word param_301;        // ?
    word param_302;        // ?
//    word param_304;        // ?
//    word param_800;        // ?
//    word param_900;        // ?
} Hitachi_status;

Here the mode list and fan_list table.

const char mode_list[10][10] = {
    {"off"},        //0x0000
    {"heat"},        //0x0010
    {"dry"},        //0x0020
    {"-"},
    {"cool"},        //0x0040
    {"fan_only"},            //0x0050
//    {"auto"},        //0x8000
    {"auto_heat"},            //0x8010
    {"auto_dry"},            //0x8020
    {"-"},
    {"auto_cool"},            //0x8040
};
const char fan_list[5][7] = {
    {"auto"},        //0x00
    {"high"},        //0x01
    {"medium"},        //0x02
    {"low"},        //0x03
    {"silent"},        //0x04
};

I add 2 timers to trigger :

- HVAC write/scan : write to HVAC if neede or ask for status

- Send MQTT to HomeAssistant

/********************* Function management *****************/ 

if (millis()-scan_timer_counter > scan_timer) {
    scan_timer_counter = millis();  
    if(things_to_write){                                            // write to HVAC id request by callback
        set_HVAC(0x0800,0x07, false);                                // Enter setup mode
        set_HVAC(address_list[1], Hitachi_status.op_mode, true);    // Op mode
        set_HVAC(0x0300,0x0000, true);                                // Unknow
        set_HVAC(address_list[3], Hitachi_status.temperature, true);// desired temp
        set_HVAC(address_list[1], Hitachi_status.op_mode, true);    // Op mode
        set_HVAC(0x0300,0x0000, true);                                // Unknow
        set_HVAC(address_list[3], Hitachi_status.temperature, true);// desired temp
        set_HVAC(address_list[2], Hitachi_status.fan_speed, false);    // fan speed
        set_HVAC(address_list[4], Hitachi_status.swing_mode, false);// swing mode
        set_HVAC(address_list[0], Hitachi_status.power, false);        // power
        set_HVAC(0x0006,0x00, false);                                // Unknow
        set_HVAC(0x0007,0x00, false);                                // Unknow
        things_to_write = false;                            // done 
        delay(500);                                            // wait a while before send status
        request_HVAC_status();                                // read status
        Send_fct();                                            // send status to HA      
    } else {                                                // read from HVAC en send if needed
        request_HVAC_status();
        if (need_to_send_MQTT) Send_fct();                    // send status to HA
        need_to_send_MQTT = false; 
    }
}

/********************** Function send **********************/ 

if (millis()-send_timer_counter > send_timer) {
    send_timer_counter = millis(); 
    Send_fct();                                            // send status to HA
}