Next.js API cannot write to local filesystem

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- [X] I have checked to see if my question is addressed in the [FAQ](https://github.com/aws-amplify/amplify-hosting/blob/master/FAQ.md).
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### App Id

dfdjxwveeb91i

### AWS Region

sa-east-1

### Amplify Hosting feature

SSR

### Describe the bug

The same as reported in #3098

Only now do I have time to retry the suggested implementation after It was said to be solved.

Unfortunately, I get the same response from Lambda@EDGE (Api execution):

```
EROFS: read-only file system, open './service-account-file.json'"
```

### Expected behavior

The same as reported in #3098

### Reproduction steps

The same as reported in #3098

### Build Settings

_No response_

### Log output

```
# Put your logs below this line

```

### Additional information

_No response_

Start by reviewing the referenced issue #3098 and its suggested implementation, then inspect how the Next.js SSR API runs in Lambda@Edge. Reproduce the EROFS error for './service-account-file.json' and document or implement a supported resolution for the local filesystem behavior.

Written by the indexing model from the issue text.