### 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_
queue_url_1 = aws_client.sqs.create_queue(QueueName=queue_name_1)["QueueUrl"]
        queue_arn_1 = arns.sqs_queue_arn(queue_name_1)
        aws_client.events.create_event_bus(Name=bus_name)
        aws_client.events.put_rule(
            Name=rule_name,
            EventBusName=bus_name,
            EventPattern=json.dumps(TEST_EVENT_PATTERN),
        )
        aws_client.events.put_targets(
            Rule=rule_name,
            EventBusName=bus_name,
            Targets=[
                {"Id": target_id, "Arn": queue_arn, "InputPath": "$.detail"},
                {
                    "Id": target_id_1,
                    "Arn": queue_arn_1,
                },
            ],
        )
        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),
                }
            ]
        )
        def get_message(queue_url):