### 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_
)
        assert "FailedEntryCount" in rs
        assert "FailedEntries" in rs
        assert rs["FailedEntryCount"] == 0
        assert rs["FailedEntries"] == []
        aws_client.events.put_events(
            Entries=[
                {
                    "EventBusName": bus_name,
                    "Source": TEST_EVENT_PATTERN["source"][0],
                    "DetailType": TEST_EVENT_PATTERN["detail-type"][0],
                    "Detail": json.dumps(EVENT_DETAIL),
                }
            ]
        )
        
        bucket_contents = aws_client.s3.list_objects(Bucket=s3_bucket)["Contents"]
        assert len(bucket_contents) == 1
        key = bucket_contents[0]["Key"]
        s3_object = aws_client.s3.get_object(Bucket=s3_bucket, Key=key)
        actual_event = json.loads(s3_object["Body"].read().decode())
        self.assert_valid_event(actual_event)
        assert actual_event["detail"] == EVENT_DETAIL
        
        aws_client.firehose.delete_delivery_stream(DeliveryStreamName=stream_name)
        
        aws_client.s3.delete_object(Bucket=s3_bucket, Key=key)
        aws_client.s3.delete_bucket(Bucket=s3_bucket)
        self.cleanup(bus_name, rule_name, target_id)
    def test_put_events_with_target_sqs_new_region(self):