### 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_
),
                snapshot.transform.jsonpath("$..detail.source-ip-address", "ip-address"),
            ]
        )
        default_bus_rule_name = f"test-default-bus-rule-{short_uid()}"
        custom_bus_rule_name = f"test-custom-bus-rule-{short_uid()}"
        default_bus_target_id = f"test-target-default-b-{short_uid()}"
        custom_bus_target_id = f"test-target-custom-b-{short_uid()}"
        custom_bus_name = f"test-eventbus-{short_uid()}"
        role = f"test-eventbus-role-{short_uid()}"
        policy_name = f"test-eventbus-role-policy-{short_uid()}"
        aws_client.s3.put_bucket_notification_configuration(
            Bucket=s3_bucket, NotificationConfiguration={"EventBridgeConfiguration": {}}
        )
        queue_url = sqs_create_queue()
        queue_arn = sqs_queue_arn(queue_url)
        custom_event_bus = events_create_event_bus(Name=custom_bus_name)
        snapshot.match("create-custom-event-bus", custom_event_bus)
        custom_event_bus_arn = custom_event_bus["EventBusArn"]
        event_bus_policy = {
            "Version": "2012-10-17",
            "Statement": [
                {"Effect": "Allow", "Action": "events:PutEvents", "Resource": custom_event_bus_arn}
            ],
        }
        role_response = create_role(
            RoleName=role, AssumeRolePolicyDocument=json.dumps(EVENT_BUS_ROLE)
        )
        role_arn = role_response["Role"]["Arn"]
        policy_arn = create_policy(