### 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_
resource_method = resource_methods.get(method, {})
    if not resource_method:
        
        
        resource_method = resource_methods.get("ANY", {}) or resource_methods.get(
            "X-AMAZON-APIGATEWAY-ANY-METHOD", {}
        )
    method_integration = resource_method.get("methodIntegration")
    if not method_integration:
        if method == "OPTIONS" and "Origin" in headers:
            
            return get_cors_response(headers)
        return make_error_response(
            "Unable to find integration for: %s %s (%s)" % (method, invocation_path, raw_path),
            404,
        )
    
    invocation_context.resource_path = extracted_path
    invocation_context.integration = method_integration
    return invoke_rest_api_integration(invocation_context)
def invoke_rest_api_integration(invocation_context: ApiInvocationContext):
    try:
        response = invoke_rest_api_integration_backend(invocation_context)
        
        
        invocation_context.response = response
        response = apply_response_parameters(invocation_context)
        return response
    except ApiGatewayIntegrationError as e:
        LOG.warning(