Fails with `private_source_authentication_failure` for a public repository

### Is there an existing issue for this?

- [x] I have searched the existing issues

### Package ecosystem

Docker

### Package manager version

_No response_

### Language version

_No response_

### Manifest location and content before the Dependabot update

Dockerfile: https://github.com/efasel/fleet-dependabot/blob/main/Dockerfile
```
FROM fleetdm/fleet:v4.64.0
```

### dependabot.yml content

https://github.com/efasel/fleet-dependabot/blob/main/.github/dependabot.yml

```
version: 2

updates:
- package-ecosystem: "docker"
directory: "/"
schedule:
interval: "weekly"
```

### Updated dependency

* from: [v4.64.0](https://hub.docker.com/layers/fleetdm/fleet/v4.64.0/images/sha256-be6d416dc209718f8c4b4328b6704880bb51039a548df09cc40736787e5bf291)
* to: [v4.64.1](https://hub.docker.com/layers/fleetdm/fleet/v4.64.1/images/sha256-7cbd70d03edad1f27221489406440e23e2bde5a7f277952b217d6a378244198a)

### What you expected to see, versus what you actually saw

I expected to get a pull request like "Bump fleetdm/fleet from v4.64.0 to v4.64.1 in /"

Instead the dependabot run failed.

* log of failing dependabot run: https://github.com/efasel/fleet-dependabot/actions/runs/13567189333/job/37922845763

[summary at the end:](https://github.com/efasel/fleet-dependabot/runs/37922845763?check_suite_focus=true#step:3:7233)
```
Dependabot encountered '1' error(s) during execution, please check the logs for more details.
+-------------------------------------------------------+
| Dependencies failed to update |
+---------------+---------------------------------------+
| fleetdm/fleet | private_source_authentication_failure |
+---------------+---------------------------------------+
Failure running container 6a1bf00a7faa552294f48f53af6a72ad7b1d1e5bddbf8d46c3bb5882faeb5efc
Cleaned up container 6a1bf00a7faa552294f48f53af6a72ad7b1d1e5bddbf8d46c3bb5882faeb5efc
proxy | 2025/02/27 13:17:24 1182/3557 calls cached (33%)
proxy | 2025/02/27 13:17:24 Posting metrics to remote API endpoint
proxy | 2025/02/27 13:17:24 Successfully posted metrics data via api client
Error: Dependabot encountered an error performing the update

Error: The updater encountered one or more errors.

For more information see: https://github.com/efasel/fleet-dependabot/network/updates/971624566 (write access to the repository is required to view the log)
🤖 ~ finished: error reported to Dependabot ~
```

### Native package manager behavior

—

### Images of the diff or a link to the PR, issue, or logs

—

### Smallest manifest that reproduces the issue

https://github.com/efasel/fleet-dependabot is a repo just to show this behaviour. It only contains this one-line Dockerfile plus the dependabot.yml for Docker.

Reproduce the failure with the linked fleet-dependabot repository, its Dockerfile, and .github/dependabot.yml. Start with the linked Actions run and Dependabot update log, then trace the Docker update path handling fleetdm/fleet and private_source_authentication_failure. Done means the public image updates from v4.64.0 to v4.64.1 and produces the expected pull request.

Written by the indexing model from the issue text.