### 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 localstack.services.apigateway.integration import (
    ApiGatewayIntegrationError,
    DynamoDBIntegration,
    HTTPIntegration,
    KinesisIntegration,
    LambdaIntegration,
    LambdaProxyIntegration,
    MockIntegration,
    S3Integration,
    SNSIntegration,
    SQSIntegration,
    StepFunctionIntegration,
)
from localstack.services.apigateway.models import ApiGatewayStore
from localstack.utils.aws import aws_stack
LOG = logging.getLogger(__name__)
class AuthorizationError(Exception):
    pass
class RequestValidator:
    __slots__ = ["context", "rest_api_container"]
    def __init__(self, context: ApiInvocationContext, store: ApiGatewayStore = None):
        self.context = context
        store = store or get_apigateway_store(
            account_id=context.account_id, region=context.region_name
        )
        if not (container := store.rest_apis.get(context.api_id)):
            
            raise NotFound()
        self.rest_api_container = container
    def is_request_valid(self) -> bool: