### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
"""
    Simple implementation around a Router to manage dynamic restapi routes (routes added by a user through the
    apigateway API).
    """
    router: Router[Handler]
    def __init__(self, router: Router[Handler]):
        self.router = router
        self.registered = False
    def register_routes(self) -> None:
        """Registers parameterized routes for API Gateway user invocations."""
        if self.registered:
            LOG.debug("Skipped API Gateway route registration (routes already registered).")
            return
        self.registered = True
        LOG.debug("Registering parameterized API Gateway routes.")
        host_pattern = "<regex('[^-]+'):api_id><regex('(-vpce-[^.]+)?'):vpce_suffix>.execute-api.<regex('.*'):server>"
        self.router.add(
            "/",
            host=host_pattern,
            endpoint=self.invoke_rest_api,
            defaults={"path": "", "stage": None},
            strict_slashes=True,
        )
        self.router.add(
            "/<stage>/",
            host=host_pattern,
            endpoint=self.invoke_rest_api,
            defaults={"path": ""},