### 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_
from datetime import datetime
from typing import Dict, List, Tuple
import pytest
from botocore.exceptions import ClientError
from pytest_httpserver import HTTPServer
from werkzeug import Request, Response
from localstack import config
from localstack.aws.api.lambda_ import Runtime
from localstack.services.events.provider import _get_events_tmp_dir
from localstack.testing.aws.util import is_aws_cloud
from localstack.utils.aws import arns, aws_stack, resources
from localstack.utils.files import load_file
from localstack.utils.strings import long_uid, short_uid, to_str
from localstack.utils.sync import poll_condition, retry
from localstack.utils.testutil import check_expected_lambda_log_events_length
from .awslambda.test_lambda import TEST_LAMBDA_PYTHON_ECHO
THIS_FOLDER = os.path.dirname(os.path.realpath(__file__))
TEST_EVENT_BUS_NAME = "command-bus-dev"
EVENT_DETAIL = {"command": "update-account", "payload": {"acc_id": "0a787ecb-4015", "sf_id": "baz"}}
TEST_EVENT_PATTERN = {
    "source": ["core.update-account-command"],
    "detail-type": ["core.update-account-command"],
    "detail": {"command": ["update-account"]},
}
API_DESTINATION_AUTHS = [
    {
        "type": "BASIC",
        "key": "BasicAuthParameters",
        "parameters": {"Username": "user", "Password": "pass"},
    },