### 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_
region_name = invocation_context.region_name or aws_stack.get_region()
    client = aws_stack.connect_to_service("apigateway", region_name=region_name)
    authorizers = client.get_authorizers(restApiId=invocation_context.api_id, limit=100).get(
        "items", []
    )
    for authorizer in authorizers:
        run_authorizer(invocation_context, authorizer)
def validate_api_key(api_key: str, invocation_context: ApiInvocationContext):
    usage_plan_ids = []
    client = connect_to(
        aws_access_key_id=invocation_context.account_id, region_name=invocation_context.region_name
    ).apigateway
    usage_plans = client.get_usage_plans()
    for item in usage_plans.get("items", []):
        api_stages = item.get("apiStages", [])
        for api_stage in api_stages:
            if (
                api_stage.get("stage") == invocation_context.stage
                and api_stage.get("apiId") == invocation_context.api_id
            ):
                usage_plan_ids.append(item.get("id"))
    for usage_plan_id in usage_plan_ids:
        usage_plan_keys = client.get_usage_plan_keys(usagePlanId=usage_plan_id)
        for key in usage_plan_keys.get("items", []):
            if key.get("value") == api_key:
                
                api_key = client.get_api_key(apiKey=key.get("id"))
                return api_key.get("enabled") in ("true", True)
    return False
def is_api_key_valid(invocation_context: ApiInvocationContext) -> bool: