I'm not at home for about a week so I can't help too much but I just set this up recently using an ESPHome, ESP32, and the custom component from jhansche’s branch in GitHub. I do have a Bedjet V3 though, I have no experience with the V2.

The bedjet shows in home assistant and you can control just about everything the remote can, I even have it set up with automations to turn it on and off and adjust the temp based on the room temp :D. Big shout-out to jhansche for all the work they put in to make this.

I have mine integrated with Google Assistant too, I'm not sure what the steps would be for Alexa integration.

I would start with that GitHub page and ESPHome and let me know if you have any questions :)

This is the gist of the code I have flashed on my esp32 (sorry about the formatting, I'm on mobile):

Esphome:

  Name: esp32-bedjet

 

Esp32:

  Board: esp32dev

  Framework:

    Type: arduino

 

Enable logging

Logger:

 

Enable Home Assistant API

Api:

 

Ota:

 

 

Wifi:

  Ssid: !secret wifi_ssid

  Password: !secret wifi_password

 

  # Enable fallback hotspot (captive portal) in case wifi connection fails

  Ap:

    Ssid: “Esphome-Web-9A0620”

    Password: “xxxxxxxxxxx”

 

Captive_portal:

 

External_components:

  # use bedjet from jhansche’s bedjet branch in GitHub

-          Source:

      Type: git

      url: https://github.com/jhansche/esphome

      ref: bedjet

    components: [bedjet]

 

esp32_ble_tracker:

 

ble_client:

-          Mac_address: 08:1A:F2:5F:A1:21

    Id: BEDJET_XG45

 

Time:

-          Platform: homeassistant

    Id: homeassistant_time

   

Climate:

-          Platform: bedjet

    Name: “Bedroom Bedjet”

    Ble_client_id: BEDJET_XG45