As I mentioned, my first step is to ignore the RS485 stuff and get the hot button mechanism working with this new hardware setup. Because I used ESPHome, things are mostly abstracted. Here are the changes I made based on testing the new assembly:

• Changed the node name to "hotbuttonrelay-al" ("al" for Atom Lite) to prevent a collision with the existing hardware.
• Modify the board definition and the referenced GPIO pins for the button and the relay trigger.
• I don't have an overly convenient way to wire in the BME280 climate sensor, so I just deleted that definition along with the I2C bus. (I have plenty of temperature sensors around the house, so no biggie).
• The Atom Lite doesn't have the usual on-board LED of typical ESP8266 or ESP32 development boards. Instead, it has a more sophisticated RGB LED. I redefined that as an ESPHome RGB light object using the fastled platform. (The compilation of the fastled libraries always gives a pile of warnings, but they don't seem harmful so far.)
• The mini relay board does not have an LED for visual feedback, so I configured the Atom Light LED to turn on and off colored green with the relay activation. Since the Atom Light is the same component used for the remote hot buttons, I cloned that bit of YAML to have it use the same hot and cold lighting effects. The relay activation uses 500ms. All the other handshaking and activity happens faster than that, so I put in a 600ms delay before the hot lighting effect starts. That lets me see the brief green signal for the relay activation.

Here's the YAML so far:

# https://hackaday.io/project/202744-calling-for-hot-water-the-recall

substitutions:
  node_name: hotbuttonrelay-al
  RELAY:  'GPIO26'
  LED:    'GPIO27'
  BUTTON: 'GPIO39'
  log_level: 'DEBUG'

esphome:
  name: ${node_name}
  on_boot:
    then:
      # delay to allow other boot time stuff to settle down
      - delay: 5s
      - script.execute: set_led_cold

esp32:
  board: pico32

wifi:
  ssid:      !secret wifi_ssid
  id:        !secret wifi_ssid
  password:  !secret wifi_password
  power_save_mode: high
  fast_connect: on
  manual_ip:
    static_ip: !secret hotbuttonrelay-al_ip
    gateway:   !secret wifi_gateway
    subnet:    !secret wifi_subnet
    dns1:      !secret wifi_dns1
    dns2:      !secret wifi_dns2

logger:
   level: ${log_level}
api:
  encryption:
    key: !secret hotbuttonrelay-al_apikey
  reboot_timeout: 60min
  services:
    - service: set_led_hot
      then:
        - logger.log:
            tag: 'hotbutton'
            level: INFO
            format: "service: set_led_hot"
        - script.execute: set_led_hot
    - service: set_led_cold
      then:
        - logger.log:
            tag: 'hotbutton'
            level: INFO
            format: "service: set_led_cold"
        - script.execute: set_led_cold
    - service: set_led_off
      then:
        - logger.log:
            tag: 'hotbutton'
            level: INFO
            format: "service: set_led_off"
        - script.execute: set_led_off

ota:
  platform: esphome
  password: !secret ota_password

switch:
  - platform: restart
    name: "${node_name} Reboot"

  - platform: gpio
    id: i_relay
    name: '${node_name} relay'
    pin: '${RELAY}'
    restore_mode: ALWAYS_OFF
    icon: mdi:hot-tub
    on_turn_on:
      - light.turn_on:
          id: bright_light
          red: 0%
          green: 100%
          blue: 0%
      - delay: 500ms
      - switch.turn_off: i_relay
      - light.turn_off:
          id: bright_light

# Local test button. This goes through the relay logic, 
# so it's not an analog press of the water heater button.
binary_sensor:
  - platform: gpio
    name: "${node_name} Button"
    id: i_button
    pin:
      number: '${BUTTON}'
      inverted: true
      mode:
        input: true
    filters:
      - delayed_on: 10ms
      - delayed_off: 10ms

light:
  - platform: fastled_clockless
    id: bright_light
    name: ${node_name} LED
    chipset: SK6812
    pin: '${LED}'
    num_leds: 1
    rgb_order: GRB # required
    default_transition_length: 0s
    icon: mdi:led-on
    effects:
      - pulse:
          name: fast_pulse
          transition_length: 0.25s
          update_interval: 0.25s
          min_brightness: 20%
          max_brightness: 99%

script:
  - id: set_led_hot
    then:
      - delay: 600ms  # give the relay blink a chance to show
      - light.turn_on:
          id: bright_light
          red: 100%
          green: 60%
          blue: 0%
          effect: fast_pulse

  - id: set_led_cold
    then:
      - light.turn_on:
          id: bright_light
          color_brightness: 20%
          red: 0%
          green: 0%
          blue: 100%
          effect: none

  - id: set_led_off
    then:
      - light.turn_on:
          id: bright_light
          color_brightness: 0%
          red: 0%
          green: 0%
          blue: 0%
          effect: none

A side note for you young tinkerers: The first time I uploaded the firmware, I forgot to change the configured static IP. That meant an IP address collision with the existing hardware. In short order, both devices were knocked off the air. Don't be like me!

Without physically hooking this new assembly to the water heater, I tested it by configuring it into my Home Assistant automation that glues everything together. Everywhere that I have a reference to the old equipment in the automation and script, I made a cloned reference that instead references the new equipment. That means that the automation activates both the old and new relays. I can hear the tiny click when the new relay is activated.