[Bug]: Playback state stays "playing" after pausing when using a Bluetooth speaker

### Describe the bug

With Alexa Media Player 5.15.7, I play Amazon Music or radio on an Echo connected to an external Bluetooth speaker. When I say "Alexa, pause", the audio stops, but Home Assistant keeps showing "playing" for at least 15 minutes.

Calling homeassistant.update_entity does not fix it. Disconnecting Bluetooth immediately corrects the state. Pause detection worked in my tests with version 5.15.2.

Could this be related to [PR #3462](https://github.com/alandtse/alexa_media_player/pull/3462)? It introduced a Bluetooth state override that might also affect playback from an Echo to an external speaker.

### To Reproduce

1. Connect an external Bluetooth speaker to an Echo.
2. Play Amazon Music or radio on the Echo.
3. Say "Alexa, pause" and confirm that the audio stops.
4. Check the Echo's state in Home Assistant. It remains "playing" for at least 15 minutes.
5. Call homeassistant.update_entity. The state still does not change.
6. Disconnect Bluetooth. The state immediately corrects itself.

### Expected behavior

Home Assistant should show "paused" when playback pauses, without needing to disconnect Bluetooth. This worked in my tests with version 5.15.2.

### Screenshots

_No response_

### Home Assistant Version

2025.11.3

### Alexa Media Player Version

5.15.7

### Alexa Media Player API library Version

Not verified

### Have you entered a 52-character key for Amazon 2SV via Authenticator App login?

Yes

### Is the HA core integration Alexa Devices installed?

No

### Amazon Domain

amazon.de (Germany)

### Amazon Domain (if Other)

_No response_

### Debug Logs (alexa_media & alexapy)

```text

```

### Additional context

_No response_