Honeywell Lyric T9/T10 RCHTSENSOR entities remain Unavailable despite valid room sensor data in API response

### The problem

### The problem

I have a Honeywell T9/T10 thermostat with two paired Honeywell Home RCHTSENSOR remote room sensors.

The main thermostat works correctly in Home Assistant.

The two remote sensor devices are also discovered and created correctly, but all of their sensor entities remain Unavailable.

Affected entities on both remote sensors:
- Room average temperature
- Room humidity
- Room temperature

This appears related to #177010 / #177022, but the behavior is different from the original issue.

In my case, the RCHTSENSOR devices and entities are now being created, but the entities never receive state values.

### What I expected to happen

The remote room sensor entities should show the temperature and humidity values returned by the Honeywell API.

### What actually happens

The devices appear correctly in Home Assistant:

- Bedroom Sensor (RCHTSENSOR)
- Basement Sensor (RCHTSENSOR)

But all three entities on each device remain Unavailable.

The main T9/T10 thermostat entities work normally.

### Important diagnostic detail

With Lyric debug logging enabled, the Honeywell API response clearly contains valid data for both remote sensors.

For example, the API response contains room data similar to:

Master Bedroom:
- roomAvgTemp: 73.492
- roomAvgHumidity: 67
- accessory temperature: 73.5
- status: Ok

Basement:
- roomAvgTemp: 72.391
- roomAvgHumidity: 49
- accessory temperature: 72.4
- status: Ok

So the remote sensor data is being returned successfully by Honeywell, but Home Assistant does not appear to populate that data into the RCHTSENSOR entities.

The Lyric coordinator also reports successful updates.

### Related issues

This appears related to:

- #177010
- #177022

However, #177022 appears to have fixed discovery/creation of the T9/T10 room sensor entities.

My current issue is that the devices/entities are created, but their states remain Unavailable despite valid data being present in the API response.

### Environment

Home Assistant Core: 2026.9.1

Thermostat:
- Honeywell T9/T10

Remote sensors:
- 2x Honeywell Home RCHTSENSOR

### Additional information

I enabled debug logging for the Lyric integration and can provide additional sanitized logs if needed.

There are occasional authentication refresh messages such as:

[homeassistant.components.lyric.coordinator] Authentication failed. Attempting to refresh token

However, the coordinator continues fetching successfully afterward, and the API responses still contain valid remote sensor data.

I can also provide screenshots showing:
- the RCHTSENSOR devices created correctly
- all remote sensor entities showing Unavailable
- the main thermostat working normally

### What version of Home Assistant Core has the issue?

core-2026.9.1

### What was the last working version of Home Assistant Core?

core-2026.8.1 ?

### What type of installation are you running?

Home Assistant OS

### Integration causing the issue

Honeywell Lyric

### Link to integration documentation on our website

https://www.home-assistant.io/integrations/lyric

### Diagnostics information

_No response_

### Example YAML snippet

```yaml
n/a
```

### Anything in the logs that might be useful for us?

```txt
2026-09-06 11:04:59.576 DEBUG (MainThread) [aiolyric]
Honeywell API response contained room/remote sensor data:

Living Room:
roomAvgTemp: 72.531
roomAvgHumidity: 73.0
accessory:
type: Thermostat
temperature: 70.0
status: Ok

Master Bedroom:
roomAvgTemp: 73.492
roomAvgHumidity: 67.0
accessory:
type: IndoorAirSensor
temperature: 73.5
status: Ok

Basement:
roomAvgTemp: 72.391
roomAvgHumidity: 49.0
accessory:
type: IndoorAirSensor
temperature: 72.4
status: Ok

2026-09-06 11:04:59.678 DEBUG (MainThread)
[homeassistant.components.lyric.coordinator]
Finished fetching lyric_coordinator data in 0.502 seconds (success: True)

2026-09-06 11:05:00.991 DEBUG (MainThread) [aiolyric]
currentPriority again contained:
Master Bedroom:
roomAvgTemp: 73.492
roomAvgHumidity: 67.0
IndoorAirSensor temperature: 73.5
status: Ok

Basement:
roomAvgTemp: 72.391
roomAvgHumidity: 49.0
IndoorAirSensor temperature: 72.4
status: Ok

2026-09-06 11:05:00.992 DEBUG (MainThread)
[homeassistant.components.lyric.coordinator]
Finished fetching lyric_coordinator data in 1.305 seconds (success: True)
```

### Additional information