Customizing native/tuya "climate" functionality

I have some Tuya-based thermostats, modded with esp chip and installed ESPHome.

Problem

The Tuya Climate component exposes data points, but in Home Assistant it only provides 2 modes: heat (on) and off. No custom operation modes or presets.

But the thermostat provides them and I have them as switches & selects in ESPHome:

What others do

for example wthermostatbeca software installed onto the ESP can automatically create some modes or presets that are available on the device.

My current solution in ESPHome

I made the climate:tuya as internal (hidden), a template thermostat climate with presets and a lot of automations that would turn on/off the switches, pick correct options from the internal input selects, etc.

Question

The only thing missing is creating custom icons that would be visible in the UI.

This seems impossible currently. But somehow WthermostatBeca software (using MQTT integration) is capable of doing it.

To get the Auto icon (Thermostat Climate Controller — ESPHome), i the thermostat would also need a cooling action, which creates a cool icon and dual gauges… But this thermostat is only for heating.

i am aware of lovelace simple-thermostat and all plugins, i just would like to provide a native way from esphome to home-assistant to provide a mode like Auto or Eco that would work natively like with other thermostats.